Since the OP asked for a discussion of features, I provide one by way of an explanation of Christian Remling's counterexample:

Holder's inequality states that $$ | \int fg | \leq \| f\|_p \|g\|_q $$ if $p^{-1} + q^{-1} = 1$, and the dual characterization of $L^p$ states that $$ f \in L^p \iff \exists M \forall g\in L^q, \int fg \leq M \|g\|_q $$

Your question can be interpreted as "does the dual characterization of $L^p$ work if, instead of all $g$ in $L^q$, we only look at those $g$s which are characteristic functions of balls (here we use that for $g$ the characteristic function of $B(x,r)$ in $\mathbb{R}^n$, its $L^q$ norm is $\approx r^{n/q}$).

While one may expect that the set of characteristic functions of balls form a nice "basis" for $L^q$, the problem however is that we don't see some convexity properties: knowing that the bounds hold for $g_1$ and $g_2$ only gives

$$ \int f (g_1 + g_2) \leq M (\|g_1\|_q + \|g_2 \|_q) $$

the right hand side of which is not $\leq \|g_1 + g_2\|_q$. In Christian Remling's answer his $f$ is built out of functions $$ \chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})} $$ We have for this function, in $\mathbb{R}^n$ $$ \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})}\|_q \approx 2^{-nk/q - k^2/q} \ll 2^{-nk/q} \approx \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})}\|_q + \|\chi_{B(0,2^{-k})}\|_q $$ These functions are, in some sense, "orthogonal" to the characteristic functions of balls, and so their largeness are not detectable by integration against balls.

Edit: since there are some questions about the actual construction of the counterexample, here's an outline of how one can construct one.

1. We will work in $\mathbb{R}$ for simplicity. You can do the same on $\mathbb{R}^2$ using balls instead of intervals.
2. We will study $f = |\nabla u|^2$ to simplify notation. Throughout $f$ will be non-negative. The requirements are now $$ \int_{I} f \leq r^{1-\delta} \tag{req}$$ for all $r\in (0,1]$, and $I$ an interval of width $r$.
3. Given an interval $I$ with with $|I| \leq 1$. Let $J$ be the interval with the same center as $I$, but width $|J| = \frac1{3^{1/(1-\delta)}} |I|^{1/(1-\delta)} \leq\frac13 |I|$. Denote by $$ f_I(x) = \begin{cases} \frac{1}{|J|^\delta} & x\in J \\ 0 & x\not\in J \end{cases} $$ One can check that $f_I$ satisfies the requirement (req), as given any interval $I'$ we have $$\int_{I'} f_I \leq |J|^{-\delta} \min(|I'|,|J|) \leq |I'|^{1-\delta}$$
4. Now take $\mathscr{I}$ a collection of disjoint intervals, then $f = \sum_{I\in \mathscr{I}} f_I$ is a function that satisfies (req): Given an arbitrary interval $I'$, if it only intersects one of the $J$s from our construction, then by the previous part we have that the inequality holds. When $I'$ spans multiple $J$s, let $\mathscr{I}'$ be the set of the corresponding $I$s, we have $$ \int_{I'} f \leq \sum_{I\in \mathscr{I}'} |J|^{1-\delta} = \sum_{I\in \mathscr{I}'} \frac13 |I| $$ But the length of $I'$ must be at least $\frac13 \sum_{I\in \mathscr{I}'} |I|$ as the $I$ in $\mathscr{I}'$ are disjoint intervals, and the $J$s are in their centers.
5. The $L^1$ norm of $f_I$ is exactly $\frac13 |I|$. For $p > 1$, the $L^p$ norm of $f_I$ is $$ \| f_I\|_{p}^p = \left(\frac13|I|\right)^{\frac{1-\delta p }{1-\delta}} $$ Note that the exponent is $< 1$.
6. Finally, take $w_n$ a sequence of numbers, each $\leq \frac13$, such that $\sum w_n$ converges but $\sum (w_n)^{\alpha}$ diverges for every $\alpha < 1$ (so something like $w_n = ( n \ln(n)^2)^{-1}$). Choose a disjoint family of intervals $I_n$ such that $|I_n| = w_n$, and setting $f = \sum f_{I_n}$. This gives an example of an $L^1$ function satisfying condition (req) that is not in any $L^p$ for $p > 1$.

Since the OP asked for a discussion of features, I provide one by way of an explanation of Christian Remling's counterexample:

Holder's inequality states that $$ | \int fg | \leq \| f\|_p \|g\|_q $$ if $p^{-1} + q^{-1} = 1$, and the dual characterization of $L^p$ states that $$ f \in L^p \iff \exists M \forall g\in L^q, \int fg \leq M \|g\|_q $$

Your question can be interpreted as "does the dual characterization of $L^p$ work if, instead of all $g$ in $L^q$, we only look at those $g$s which are characteristic functions of balls (here we use that for $g$ the characteristic function of $B(x,r)$ in $\mathbb{R}^n$, its $L^q$ norm is $\approx r^{n/q}$).

While one may expect that the set of characteristic functions of balls form a nice "basis" for $L^q$, the problem however is that we don't see some convexity properties: knowing that the bounds hold for $g_1$ and $g_2$ only gives

$$ \int f (g_1 + g_2) \leq M (\|g_1\|_q + \|g_2 \|_q) $$

the right hand side of which is not $\leq \|g_1 + g_2\|_q$. In Christian Remling's answer his $f$ is built out of functions $$ \chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})} $$ We have for this function, in $\mathbb{R}^n$ $$ \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})}\|_q \approx 2^{-nk/q - k^2/q} \ll 2^{-nk/q} \approx \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})}\|_q + \|\chi_{B(0,2^{-k})}\|_q $$ These functions are, in some sense, "orthogonal" to the characteristic functions of balls, and so their largeness are not detectable by integration against balls.

Edit: since there are some questions about the actual construction of the counterexample, here's an outline of how one can construct one.

1. We will work in $\mathbb{R}$ for simplicity. You can do the same on $\mathbb{R}^2$ using balls instead of intervals.
2. We will study $f = |\nabla u|^2$ to simplify notation. Throughout $f$ will be non-negative. The requirements are now $$ \int_{I} f \leq r^{1-\delta} \tag{req}$$ for all $r\in (0,1]$, and $I$ an interval of width $r$.
3. Given an interval $I$ with with $|I| \leq 1$. Let $J$ be the interval with the same center as $I$, but width $|J| = \frac1{3^{1/(1-\delta)}} |I|^{1/(1-\delta)} \leq\frac13 |I|$. Observe that as $I$ shrinks in length, the interval $J$ shrinks faster. Denote by $$ f_I(x) = \begin{cases} \frac{1}{|J|^\delta} & x\in J \\ 0 & x\not\in J \end{cases} $$ One can check that $f_I$ satisfies the requirement (req), as given any interval $I'$ we have $$\int_{I'} f_I \leq |J|^{-\delta} \min(|I'|,|J|) \leq |I'|^{1-\delta}$$
4. Now take $\mathscr{I}$ a collection of disjoint intervals, then $f = \sum_{I\in \mathscr{I}} f_I$ is a function that satisfies (req): Given an arbitrary interval $I'$, if it only intersects one of the $J$s from our construction, then by the previous part we have that the inequality holds. When $I'$ spans multiple $J$s, let $\mathscr{I}'$ be the set of the corresponding $I$s, we have $$ \int_{I'} f \leq \sum_{I\in \mathscr{I}'} |J|^{1-\delta} = \sum_{I\in \mathscr{I}'} \frac13 |I| $$ But the length of $I'$ must be at least $\frac13 \sum_{I\in \mathscr{I}'} |I|$ as the $I$ in $\mathscr{I}'$ are disjoint intervals, and the $J$s are in their centers.
5. The $L^1$ norm of $f_I$ is exactly $\frac13 |I|$. For $p > 1$, the $L^p$ norm of $f_I$ is $$ \| f_I\|_{p}^p = \left(\frac13|I|\right)^{\frac{1-\delta p }{1-\delta}} $$ Note that the exponent is $< 1$.
6. Finally, take $w_n$ a sequence of numbers, each $\leq \frac13$, such that $\sum w_n$ converges but $\sum (w_n)^{\alpha}$ diverges for every $\alpha < 1$ (so something like $w_n = ( n \ln(n)^2)^{-1}$). Choose a disjoint family of intervals $I_n$ such that $|I_n| = w_n$, and setting $f = \sum f_{I_n}$. This gives an example of an $L^1$ function satisfying condition (req) that is not in any $L^p$ for $p > 1$.

Since the OP asked for a discussion of features, I provide one by way of an explanation of Christian Remling's counterexample:

Holder's inequality states that $$ | \int fg | \leq \| f\|_p \|g\|_q $$ if $p^{-1} + q^{-1} = 1$, and the dual characterization of $L^p$ states that $$ f \in L^p \iff \exists M \forall g\in L^q, \int fg \leq M \|g\|_q $$

Your question can be interpreted as "does the dual characterization of $L^p$ work if, instead of all $g$ in $L^q$, we only look at those $g$s which are characteristic functions of balls (here we use that for $g$ the characteristic function of $B(x,r)$ in $\mathbb{R}^n$, its $L^q$ norm is $\approx r^{n/q}$).

While one may expect that the set of characteristic functions of balls form a nice "basis" for $L^q$, the problem however is that we don't see some convexity properties: knowing that the bounds hold for $g_1$ and $g_2$ only gives

$$ \int f (g_1 + g_2) \leq M (\|g_1\|_q + \|g_2 \|_q) $$

the right hand side of which is not $\leq \|g_1 + g_2\|_q$. In Christian Remling's answer his $f$ is built out of functions $$ \chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})} $$ We have for this function, in $\mathbb{R}^n$ $$ \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})}\|_q \approx 2^{-nk/q - k^2/q} \ll 2^{-nk/q} \approx \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})}\|_q + \|\chi_{B(0,2^{-k})}\|_q $$ These functions are, in some sense, "orthogonal" to the characteristic functions of balls, and so their largeness are not detectable by integration against balls.

Since the OP asked for a discussion of features, I provide one by way of an explanation of Christian Remling's counterexample:

Holder's inequality states that $$ | \int fg | \leq \| f\|_p \|g\|_q $$ if $p^{-1} + q^{-1} = 1$, and the dual characterization of $L^p$ states that $$ f \in L^p \iff \exists M \forall g\in L^q, \int fg \leq M \|g\|_q $$

Your question can be interpreted as "does the dual characterization of $L^p$ work if, instead of all $g$ in $L^q$, we only look at those $g$s which are characteristic functions of balls (here we use that for $g$ the characteristic function of $B(x,r)$ in $\mathbb{R}^n$, its $L^q$ norm is $\approx r^{n/q}$).

While one may expect that the set of characteristic functions of balls form a nice "basis" for $L^q$, the problem however is that we don't see some convexity properties: knowing that the bounds hold for $g_1$ and $g_2$ only gives

$$ \int f (g_1 + g_2) \leq M (\|g_1\|_q + \|g_2 \|_q) $$

the right hand side of which is not $\leq \|g_1 + g_2\|_q$. In Christian Remling's answer his $f$ is built out of functions $$ \chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})} $$ We have for this function, in $\mathbb{R}^n$ $$ \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})} - \chi_{B(0,2^{-k})}\|_q \approx 2^{-nk/q - k^2/q} \ll 2^{-nk/q} \approx \|\chi_{B(0, 2^{-k} + 2^{-k^2 - k})}\|_q + \|\chi_{B(0,2^{-k})}\|_q $$ These functions are, in some sense, "orthogonal" to the characteristic functions of balls, and so their largeness are not detectable by integration against balls.

Edit: since there are some questions about the actual construction of the counterexample, here's an outline of how one can construct one.

1. We will work in $\mathbb{R}$ for simplicity. You can do the same on $\mathbb{R}^2$ using balls instead of intervals.
2. We will study $f = |\nabla u|^2$ to simplify notation. Throughout $f$ will be non-negative. The requirements are now $$ \int_{I} f \leq r^{1-\delta} \tag{req}$$ for all $r\in (0,1]$, and $I$ an interval of width $r$.
3. Given an interval $I$ with with $|I| \leq 1$. Let $J$ be the interval with the same center as $I$, but width $|J| = \frac1{3^{1/(1-\delta)}} |I|^{1/(1-\delta)} \leq\frac13 |I|$. Denote by $$ f_I(x) = \begin{cases} \frac{1}{|J|^\delta} & x\in J \\ 0 & x\not\in J \end{cases} $$ One can check that $f_I$ satisfies the requirement (req), as given any interval $I'$ we have $$\int_{I'} f_I \leq |J|^{-\delta} \min(|I'|,|J|) \leq |I'|^{1-\delta}$$
4. Now take $\mathscr{I}$ a collection of disjoint intervals, then $f = \sum_{I\in \mathscr{I}} f_I$ is a function that satisfies (req): Given an arbitrary interval $I'$, if it only intersects one of the $J$s from our construction, then by the previous part we have that the inequality holds. When $I'$ spans multiple $J$s, let $\mathscr{I}'$ be the set of the corresponding $I$s, we have $$ \int_{I'} f \leq \sum_{I\in \mathscr{I}'} |J|^{1-\delta} = \sum_{I\in \mathscr{I}'} \frac13 |I| $$ But the length of $I'$ must be at least $\frac13 \sum_{I\in \mathscr{I}'} |I|$ as the $I$ in $\mathscr{I}'$ are disjoint intervals, and the $J$s are in their centers.
5. The $L^1$ norm of $f_I$ is exactly $\frac13 |I|$. For $p > 1$, the $L^p$ norm of $f_I$ is $$ \| f_I\|_{p}^p = \left(\frac13|I|\right)^{\frac{1-\delta p }{1-\delta}} $$ Note that the exponent is $< 1$.
6. Finally, take $w_n$ a sequence of numbers, each $\leq \frac13$, such that $\sum w_n$ converges but $\sum (w_n)^{\alpha}$ diverges for every $\alpha < 1$ (so something like $w_n = ( n \ln(n)^2)^{-1}$). Choose a disjoint family of intervals $I_n$ such that $|I_n| = w_n$, and setting $f = \sum f_{I_n}$. This gives an example of an $L^1$ function satisfying condition (req) that is not in any $L^p$ for $p > 1$.