The nonzero frequencies contribute an error of size $O_A((\log x)^{-A})$. For these terms, break $d$ and $e$ into dyadic ranges $d \asymp D, e \asymp E$. Clearly we may assume $DE \gg x$. Also, we may assume without loss of generality that $D \leq E$, otherwise use the reciprocity relation $$\frac{\overline{d}}{e} + \frac{\overline{e}}{d}\equiv \frac{1}{de}\pmod{1}$$ to switch $d$ and $e$ in the exponential. After some work, Siegel-Walfisz and the large sieve cover the case when $D\leq E (\log x)^{-B}$, so we may assume $D \approx E$. The double sum over $d$ and $e$ may then be suitably bounded using results of Duke-Friedlander-Iwaniec on bilinear forms in Kloosterman fractions (see also work of Bettin and Chandee).

The nonzero frequencies contribute an error of size $O_A((\log x)^{-A})$. For these terms, break $d$ and $e$ into dyadic ranges $d \asymp D, e \asymp E$. Clearly we may assume $DE \gg x$. Also, we may assume without loss of generality that $E \ll D$, otherwise use the reciprocity relation $$\frac{\overline{d}}{e} + \frac{\overline{e}}{d}\equiv \frac{1}{de}\pmod{1}$$ to switch $d$ and $e$ in the exponential. After some work, Siegel-Walfisz and the large sieve cover the case when $E\leq D (\log x)^{-B}$, so we may assume $D \approx E$. The double sum over $d$ and $e$ may then be suitably bounded using results of Duke-Friedlander-Iwaniec on bilinear forms in Kloosterman fractions (see also work of Bettin and Chandee).

[Edit: March 15]

I'm adding a bit more detail, as requested. I'm assuming the smoothing $w$ is a smooth bump function, so the Fourier transform satisfies $|\widehat{w}(y)| \ll \exp(-|y|^{1/2})$, say. The arguments below might have to be adapted or substituted entirely for less smooth weights.

With $w$ as above, one finds that $$\sum_n \varphi(n)/n w(n/x) = Cx + O_A((\log x)^{-A})$$ by Mellin inversion, contour shifting, and the zero-free region for zeta.

Now, as to the correlation sum. Begin as above, and perform Poisson summation. We separate the $k=0$ term which gives the main term, and the contribution of the nonzero frequencies is $$\sum_{d\ll x}\frac{\mu(d)}{d}\sum_{\substack{e \ll x \\ (e,d)=1}}\frac{\mu(e)}{e}\frac{x}{de}\sum_{k \in \mathbb{Z}\backslash \{0\}} e\left(-k\frac{\overline{d}}{e}\right) \widehat{w}\left(k\frac{x}{de} \right).$$ Dyadically decompose $d\asymp D$ and $e \asymp E$. As mentioned above, up to changing the sign of $k$ in the exponential one may assume that $E \ll D$ by using reciprocity.

Now use the rapid decay of $\widehat{w}$ to truncate the sum on $k$. Up to a minor change in the coefficients and an acceptable error term, we have $$\frac{x}{D^2E^2}\sum_{e\asymp E} \mu(e) \sum_{\substack{d \asymp D \\ (d,e)=1}}\mu(d) \sum_{0<|k|\leq (\log x)^{10}DE/x} e\left(k\frac{\overline{d}}{e}\right)e\left(\frac{k}{de} \right) \widehat{w}\left(k\frac{x}{de} \right).$$ Clearly, we may assume $DE > x(\log x)^{-11}$, say.

We are going to use the large sieve inequality, but in order to do so we need to separate the $d$ and $e$ variables from each other inside $\widehat{w}$. Change variables in the definition of $\widehat{w}$ and interchange to obtain $$\int_{u \asymp 1} \frac{x}{D^2E^2}\sum_{0<|k|\leq (\log x)^{10}DE/x} \sum_{e \asymp E}\mu(e)w(\tfrac{eu}{E})\sum_{\substack{d \asymp D \\ (d,e)=1}}\mu(d) e\left(-k\frac{x}{dE}u \right) e\left(-k\frac{\overline{d}}{e}\right) du,$$ where this holds up to harmless changes in the $d$ and $e$ coefficients. We work uniformly in $u\asymp 1$, so we can drop the integral.

We give two arguments. The first works when $E$ is a bit smaller than $D$, and the other works when they are about the same size.

The first argument uses the large sieve. Split the sum over $d$ into arithmetic progressions modulo $e$ to separate the additive character, then apply multiplicative characters to detect the congruence condition modulo $e$. This turns the sum over $d$ into $$\frac{1}{\varphi(e)}\sum_{\chi (e)}\sum_{(a,e)=1}e\left(-k\frac{\overline{a}}{e}\right) \overline{\chi}(a)\sum_{\substack{d \asymp D}}\mu(d)\chi(d) e\left(-k\frac{x}{dE}u \right).$$ We take out the contribution of the principal character $\chi_0 \pmod{e}$. We get cancellation in the sum over $d$ using partial summation. The sum over $a\pmod{e}$ becomes a Ramanujan sum, which has size $\leq (|k|,e)$, the GCD of $|k|$ and $e$. It is easy to see that the total contribution from $\chi_0$ after summing over all the variables is $\ll_A (\log x)^{-A}$.

We have to bound the contribution of the non-principal characters: $$\frac{x}{D^2E^2}\sum_{0<|k|\leq (\log x)^{10}DE/x} \sum_{e \asymp E}\mu(e)w(\tfrac{eu}{E})\frac{1}{\varphi(e)}\sum_{\substack{\chi (e) \\ \chi \neq \chi_0}}\sum_{(a,e)=1}e\left(-k\frac{\overline{a}}{e}\right) \overline{\chi}(a)\sum_{\substack{d \asymp D}}\mu(d)\chi(d) e\left(-k\frac{x}{dE}u \right).$$ We fixed $k$, since we will not need to sum over this variable, and then apply Cauchy-Schwarz to the sum over $e$. This gives $$\sum_e \ll (\Sigma_1 \Sigma_2)^{1/2},$$ where $$\Sigma_1 = \sum_{e \asymp E} \frac{1}{\varphi(e)}\sum_{\substack{\chi (e) \\ \chi \neq \chi_0}} \left|\sum_{(a,e)=1}e\left(-k\frac{\overline{a}}{e}\right) \overline{\chi}(a) \right|^2$$ and $$\Sigma_2 = \sum_{e \asymp E} \frac{1}{\varphi(e)}\sum_{\substack{\chi (e) \\ \chi \neq \chi_0}} \left|\sum_{\substack{d \in I_D}}\mu(d)\chi(d) e\left(-k\frac{x}{dE}u \right) \right|^2.$$ We easily bound $\Sigma_1$ by using positivity to include $\chi_0$, then opening the square and using orthogonality of characters. It follows that $$\Sigma_1 \ll E^2.$$

The argument for $\Sigma_2$ is slightly more involved. First, we reduce to summing over primitive Dirichlet characters $\psi$ (cf. any standard proof of the Bombieri-Vinogradov theorem). We apply a dyadic decomposition to the conductor of the primitive characters, so we derive something like $$\Sigma_2 \ll \sum_{1 \ll R = 2^j\ll E}\sum_{f \ll E} \frac{1}{\varphi(f)}\sum_{r \asymp R} \frac{1}{\varphi(r)} \sum_{\substack{\psi( r) \\ \psi \text{ prim}}}\Big|\sum_{\substack{d \in I_D \\ (d,f)=1}}\mu(d)\psi(d) e\left(-k\frac{x}{dE}u \right) \Big|^2.$$ If $R \leq (\log x)^C$ then we use partial summation and the Siegel-Walfisz theorem to bound the sum over $d$. If $R > (\log x)^C$ we use the multiplicative large sieve inequality $$\sum_{r \asymp R} \frac{1}{\varphi(r)} \sum_{\substack{\psi( r) \\ \psi \text{ prim}}}\Big|\sum_{\substack{d \in I_D \\ (d,f)=1}}\mu(d)\psi(d) c(d)e\left(-k\frac{x}{dE}u \right) \Big|^2 \ll \frac{1}{R}\left(R^2 + D \right)D.$$ We deduce that $$\Sigma_2 \ll D^2 (\log x)^{-2A} + DE.$$

Recalling the application of Cauchy-Schwarz and the bounds for $\Sigma_1$ and $\Sigma_2$, we sum over $k$ to see that the total contribution is $$\ll (\log x)^{-A + O(1)} + (\log x)^{O(1)} \left(\frac{E}{D} \right)^{1/2}.$$ This is acceptably small if $E \leq D(\log x)^{-B}$.

Therefore, we may assume that $D x^{-o(1)} \ll E \ll D$. Since $DE \gg x^{1-o(1)}$ this implies $D \gg x^{1/2-o(1)}$. For this second argument we do not need any properties of the coefficients attached to the $d$ and $e$ variables other than that they are 1-bounded, so we wish to get a bound on $$\frac{x}{D^2E^2}\sum_{1\leq k\leq (\log x)^{10}DE/x}\Big| \mathop{\sum\sum}_{\substack{d \asymp D \\ e \asymp E \\ (d,e)=1}}\alpha_d \beta_e e \left(k \frac{\overline{d}}{e} \right)\Big|$$ for 1-bounded coefficients $\alpha_d,\beta_e$. Theorem 2 of the above-mentioned Duke-Friedlander-Iwaniec paper gives $$\Big| \mathop{\sum\sum}_{\substack{d \asymp D \\ e \asymp E \\ (d,e)=1}}\alpha_d \beta_e e \left(k \frac{\overline{d}}{e} \right)\Big| \ll (DE)^{1/2} (k+DE)^{3/8} (D+E)^{11/48}\ll D^{2-1/48+o(1)},$$ which saves $D^{1/48}$ over the trivial bound. Therefore $$\frac{x}{D^2E^2}\sum_{1\leq k\leq (\log x)^{10}DE/x}\Big| \mathop{\sum\sum}_{\substack{d \asymp D \\ e \asymp E \\ (d,e)=1}}\alpha_d \beta_e e \left(k \frac{\overline{d}}{e} \right)\Big| \ll \frac{D^{2 - \frac{1}{48} + o(1)}}{DE} \ll D^{-1/48+o(1)} \ll x^{-1/96+o(1)}.$$

If $w$ is smooth and compactly supported in $[1/2,2]$, say, then $$\sum_{n}\frac{\varphi(n)}{n}\frac{\varphi(n+1)}{n+1}w(n/x) = Cx + O(1).$$ Write $$\frac{\varphi(n)}{n} = \sum_{d\mid n} \frac{\mu(d)}{d}$$ and $$\frac{\varphi(n+1)}{n+1} = \sum_{e \mid n+1} \frac{\mu(e)}{e},$$ then interchange the orders of summation. Perform Poisson summation on the $n$ variable to turn the sum into something like $$\sum_d \frac{\mu(d)}{d} \sum_{(e,d)=1}\frac{\mu(e)}{e} \frac{x}{d^2e^2}\sum_{|k|\leq de/x} e\left(-k\frac{\overline{d}}{e}\right)\widehat{w}(kx/de),$$ where $\overline{d}$ denotes the inverse of $d$ modulo $e$. The $k=0$ term gives the main term, which is seen to be $Cx + O(1)$.

The nonzero frequencies contribute an error of size $O_A((\log x)^{-A})$. For these terms, break $d$ and $e$ into dyadic ranges $d \asymp D, e \asymp E$. Clearly we may assume $DE \gg x$. Also, we may assume without loss of generality that $D \leq E$, otherwise use the reciprocity relation $$\frac{\overline{d}}{e} + \frac{\overline{e}}{d}\equiv \frac{1}{de}\pmod{1}$$ to switch $d$ and $e$ in the exponential. After some work, Siegel-Walfisz and the large sieve cover the case when $D\leq E (\log x)^{-B}$, so we may assume $D \approx E$. The double sum over $d$ and $e$ may then be suitably bounded using results of Duke-Friedlander-Iwaniec on bilinear forms in Kloosterman fractions (see also work of Bettin and Chandee).

If $w$ is smooth and compactly supported in $[1/2,2]$, say, then $$\sum_{n}\frac{\varphi(n)}{n}\frac{\varphi(n+1)}{n+1}w(n/x) = Cx + O_A((\log x)^{-A}).$$ Write $$\frac{\varphi(n)}{n} = \sum_{d\mid n} \frac{\mu(d)}{d}$$ and $$\frac{\varphi(n+1)}{n+1} = \sum_{e \mid n+1} \frac{\mu(e)}{e},$$ then interchange the orders of summation. Perform Poisson summation on the $n$ variable to turn the sum into something like $$\sum_d \frac{\mu(d)}{d} \sum_{(e,d)=1}\frac{\mu(e)}{e} \frac{x}{d^2e^2}\sum_{|k|\leq de/x} e\left(-k\frac{\overline{d}}{e}\right)\widehat{w}(kx/de),$$ where $\overline{d}$ denotes the inverse of $d$ modulo $e$. The $k=0$ term gives the main term, which is seen to be $Cx + O_A((\log x)^{-A})$ by using the cancellation in the mobius function.

The nonzero frequencies contribute an error of size $O_A((\log x)^{-A})$. For these terms, break $d$ and $e$ into dyadic ranges $d \asymp D, e \asymp E$. Clearly we may assume $DE \gg x$. Also, we may assume without loss of generality that $D \leq E$, otherwise use the reciprocity relation $$\frac{\overline{d}}{e} + \frac{\overline{e}}{d}\equiv \frac{1}{de}\pmod{1}$$ to switch $d$ and $e$ in the exponential. After some work, Siegel-Walfisz and the large sieve cover the case when $D\leq E (\log x)^{-B}$, so we may assume $D \approx E$. The double sum over $d$ and $e$ may then be suitably bounded using results of Duke-Friedlander-Iwaniec on bilinear forms in Kloosterman fractions (see also work of Bettin and Chandee).