Aren't deep learning models themselves a case where we have hints of some deeper underlying logic to why some things are more effective than others, but we lack the mathematical tools to properly work it out for anything of practical size?

All we're able to do is apply flawed analogies, generic information theoretical models, trial and error, post-hoc rationalizations and benchmarks without really understanding why.

Doesn’t this generalize? Mathematics matters less than less as fewer people are capable of understanding it. So whatever cutting edge, deep insight about the nature of groups matters less than different equations which matters less than solving linear equations, etc.

Not really, a theorem with a hard proof can have simple but important corollaries. It's also not unthinkable that theorems exist with proofs that cannot reduce to something simple/short.