Because there are natural computational problems involving many mathematical objects, there are a bunch of implications of complexity class separations like $\mathrm{P} \neq \mathrm{NP}$. I think the first paper to investigate this idea is probably Mike Freedman's Complexity classes as mathematical axioms, which assumes a complexity class separation (namely $\mathrm{NP} \neq \mathrm{P}^{\#\mathrm{P}}$, which is stronger than $\mathrm{P} \neq \mathrm{NP}$) to prove that knots with certain properties exist.
The main idea of all these arguments is to prove an implication like "If all objects of type $T$ satisfy property $P$, then there is an efficient algorithm for a problem which we assumed has no efficient algorithm." You can then deduce the existence of objects of type $T$ which satisfy property $\neg P$. (Here the meaning of "efficient" depends on the class separation you assume.)
The exact thing Freedman proves is a little esoteric, so let me give two other examples that have a somewhat similar flavor.
The systole of a metric manifold is the length of the shortest non-contractible loop on it (I'll also use the word for the shortest loop itself). Probably the first manifold whose systole you'd try to understand is a flat torus $\mathbb{T}^d = \mathbb{R}^d / \Lambda$ for some lattice $\Lambda = \langle v_1, \dots, v_d \rangle$, since these are pretty much the simplest metric manifolds.
One natural thing might be to try to say something about the word length of the systole $\gamma$ when considered as an element of $\pi_1(\mathbb{T}^d)$ equipped with the generating set $\{ v_1^*, \dots, v_d^* \}$ where $v_i^*$ is the loop in $\mathbb{T}^d$ naturally associated to $v_i$. For example, maybe the systole always has a relatively short word expressing it. Say, maybe we can always write $\gamma =_{\pi_1(\mathbb{T}^d)} \sum_i n_i v_i^*$ with $\sum_i |n_i| < \sqrt{d}$ or something.
It turns out that a modest strengthening of $\mathrm{P} \neq \mathrm{NP}$ actually rules this out. Specifically, assuming that $\mathrm{NP}$-hard problems do not have time $2^{o(n)}$ bounded-error probabilstic algorithms, we can prove the following: For any $\ell(d) = o(d / \log d)$, there exist infinitely many $d$ and $\Lambda=(v_1, \dots, v_d)$ such that every systolic loop $\gamma$ on the torus $\mathbb{R}^d / \Lambda$ has word length at least $\ell(n)$ in the generating set $\{ v_1^*, \dots, v_d^* \}$.
The idea of the argument is just that such a bound would imply a sub-exponential time algorithm for the $\mathrm{NP}$-hard problem of computing the systole: namely just enumerate all possible words in the generators of the given word length and pick the one which is represented by the shortest loop (this is easy to compute).
You can use a similar argument also to show that faithful representations of $S_n$ have dimension at least $n^{\varepsilon}$ for some $\varepsilon > 0$ (assuming some version of the strong exponential time hypothesis.) The basic idea is given here -- the argument is very simple.
These arguments I think give some idea of why proving class separations should be hard. They immediately imply that mathematical objects of all kinds must have certain kinds of complexity, or else they could be used to give algorithms contradicting the class separations. So, the class separation simultaneously demonstrates the existence of such complexity in all such mathematical objects at once.
Bonus: Tim Roughgarden and Inbal Talgam-Cohen have some writing along these lines as well showing that class separations imply markets in which certain kinds of equilibria do not exist.
