There is probably no single proof that would provide a rigorous justification of the OP's principle in all cases. Moreover, without specifying more clearly what is meant by a 'natural map', the principle itself turns out not to hold in general.

For example, every smooth (complex-valued) function $f$ on the unit circle $S^1\subset\mathbb{C}$ can be written uniquely in the form
$$
f(e^{i\theta}) = f_0(e^{2i\theta}) + e^{i\theta}\,f_1(e^{2i\theta}),
$$
and the (natural?) 'even-odd' decomposition mapping $f(e^{i\theta})\mapsto \bigl(f_0(e^{i\theta}),f_1(e^{i\theta})\bigr)\ $ is one-to-one and onto.

As a (perhaps) more serious example, by the Nash-Kuiper $C^1$-isometric embedding theorem, any smooth Riemannian metric $g$ in dimension $n$ can be locally isometrically embedded into $\mathbb{R}^{n+1}$ by a $C^1$-mapping. Since metrics in dimension $n$ depend on $\tfrac12n(n{+}1)$ functions of $n$ variables while maps into $\mathbb{R}^{n+1}$ depend on $n{+}1$ functions of $n$ variables, this violates your heuristic principle when $n>2$. (To be sure, the Nash-Kuiper theorem was greeted with astonishment when it first appeared in 1954.)

It would be hard to argue that the mapping $\Phi(f) = \mathrm{d}f\cdot\mathrm{d}f = g$, where $f:M^n\to\mathbb{R}^{n+1}$ and $g$ is a metric on $M$ is not natural, but, of course, one can object that the differing degrees of differentiability on $f$ and $g$ should be taken into account in any careful formulation of a heuristic principle along the lines that the OP wants.

Now, there *is* a wide class of equations that includes the OP's Examples 1, 2, and 3 and explains the failure of surjectivity of each of them, namely the class of *smooth nonlinear differential operators*, $\Phi:C^\infty(E)\to C^\infty(F)$, where $E$ and $F$ are bundles over a common base manifold $M$.

To formulate this notion precisely, recall that, given a smooth bundle $E\to M$ of fiber rank $p$, say, and where $M$ is a smooth manifold of dimension $n$, one has the bundle $J^k(E)\to M$ of $k$-jets of sections of $E$, which is a smooth bundle of fiber rank $p{{n+k}\choose n}$.

Given another smooth bundle $F\to M$ of fiber rank $q$, say, a (smooth) nonlinear differential operator of order at most $s$, say $\Phi:C^\infty(E)\to C^\infty(F)$
is a mapping of the form $\Phi(u) = \Phi^s\bigl(j^s(u)\bigr)$ for all $u\in C^\infty(E)$
where $\Phi^s: J^s(E)\to F$ is a smooth bundle mapping. (In practice, one often only has $\Phi^s$ defined on an open subbundle $A^s\subset J^s(E)$, in which case one says that a section $u:M\to E$ is *$A^s$-admissible* if its associated $s$-jet section $j^s(u):M\to J^s(E)$ has its image lying in $A^s$. Then, one only gets a mapping $\Phi:C^\infty(E,A)\to C^\infty(F)$, where $C^\infty(E,A)\subset C^\infty(E)$ is the subset of $A^s$-admissible sections. The reader can deal with the details of that situation.)

In this case, the OP's heuristic principle would suggest that $\Phi$ cannot be surjective, even locally, if the rank of $F$ is greater than the rank of $E$. Indeed, this turns out to be the case, and can be proved rigorously by the argument below.

First, though, note that the OP's Example 1 is a nonlinear differential operator of order $1$, where $E = M\times \mathbb{R}^{n+1}$ and $F = S^2(T^*M)$ and $\Phi^1(u) = \mathrm{d}u\cdot\mathrm{d}u$. Here, $(p,q) = \bigl(n{+}1, {{n+1}\choose2}\bigr)$.
Similarly, the OP's Examples 2 and 3 are nonlinear differential operators of order $2$, with
$(p,q) = \bigl({{n+1}\choose2}, \frac{n^2(n^2-1)}{12}\bigr)$ in the case of Example 2, and
$(p,q) = \bigl(1, \frac{n}2\bigr)$ in the case of Example 3 (here, the underlying manifold is $T\mathbb{R}^{n/2} = \mathbb{R^n}$, on which the Lagrangian for curves would be defined).

To prove the promised nonsurjectivity, suppose given such a smooth differential operator. One then has canonical *prolongations* $\Phi^k:J^k(E)\to J^{k-s}(F)$ for $k\ge s$, which are defined by the property that
$$
\Phi^k\bigl(j^k(u)\bigr) = j^{k-s}\bigl(\Phi^s(j^s(u))\bigr)
$$
for all $k\ge s$ and all (local) sections $u\in C^\infty(E)$.

*The crucial consequence of this equation is that the $(k{-}s)$-jets of sections $v:M\to F$ that are of the form $v = \Phi(u) = \Phi^s(j^s(u))$ for some $u\in C^\infty(E)$ have to lie in the image of $\Phi^k$.*

Now, if $p<q$, then for all $k$ sufficiently large, one has
$$
\dim J^k(E) = n + p{{n+k}\choose n} < n+ q{{n+k-s}\choose n} = \dim J^{k-s}(F).
$$
Thus, the prolongation map $\Phi^k:J^k(E)\to J^{k-s}(F)$ cannot be surjective for $k$ sufficiently large, and this implies that the equation $v = \Phi^s(j^s(u))$ has no solution, even locally, for the generic section $v:M\to F$, i.e., $\Phi:C^\infty(E)\to C^\infty(F)$ is not surjective.

Finally, note that this argument heavily uses the assumption of infinite differentiability. This is not surprising because, as the Nash-Kuiper theorem shows, this non-surjectivity can indeed fail spectacularly without such assumptions.