It appears that a similar issue crops up in the interpretation of classical mechanics.

That is, suppose we have our laws of classical mechanics. Hence, we have a nice way of propagating states along time. Now we want to understand thermodynamics and the arrow of time. Maths tell us that, if we slice-and-dice microstates and macrostates in a certain way, then thermodynamics is correct. In a certain way means "any non-crazy way with respect to Liouville measure" and the entire thing only holds "for sufficiently chaotic/ergodic systems".

We observe (empirically / subjectively) that thermodynamics holds, and we have an arrow of time. In order to explain this observation, we need an additional axiom:

(A) The universe looks like it has Liouville-random but phantastically low-entropy initial state in the far past. Liouville-random means something like "absolutely continuous wrt Liouville-measure".

We could split that into

(A1) Thermodynamics makes sense. The universe cares about Liouville-measure.
(A2) With respect to Liouville-measure, we have a low-entropy state at some point in time (hence we can define "past (noun): In the time-direction of the designated point in time with low entropy").

In the overwhelming majority of such universes (classic trajectories), observers see things that are compatible with both thermodynamics and an arrow of time (and there is a possibility for conscious overservers to evolve).

Mathematically, there are many ways of measuring phase-space volume, in a way that is invariant under time-evolution. Assuming ergodicity/chaoticity/mixing, there is only a single such way that has bounded density function with respect to "ordinary n-dim volume" (SRB-measure), and all these converge to Liouville. But it would have been mathematically conceivable to take the same evolution equations, and use a different measure that is supported on weird subsets of low fractal dimension, and end up with different thermodynamics. Hence, we need an extra axiom to separate observed reality from mathematical possibility, and single out Liouville / ordinary volume.

Nobody takes issue with accepting this axiom. So why do people have issues with accepting the Born rule as an axiom?

It is clear that we must add axioms to the evolution equations to explain that we care about squared-amplitude and that we have an arrow of time. Sure, you can try to prove theorems showing that alternatives to the Born rule are crazy-weird, and make the axiom weaker ("if we care about anything at all, and the thing we care about is not batshit insane then the Born rule holds"). For example, MWI tries to replace copenhagen "if humans do experiments, use squared-amplitude" with axiom "the universe cares about squared-amplitudes in a non-crazy way" plus pseudo-theorem "human-scale experiments can be approximated by Born rule". Certainly more elegant, because the universe doesn't need to care about squishy emergent approximate concepts like "humans" anymore. But it is the same axiom in the end, modulo some plausible not-quite-proven maths for the pseudo-theorem.

Not-quite-proven even in the classical case: Formally proving unique ergodicity / chaoticity / fast mixing is far beyond us for almost all non-trivial systems. Just like e.g. complexity theory (we don't even have a formal proof of P!=NP, so the name of the game is "mathematical plausibility" that is spot-checked by tiny little theorems).