"Our puny brains" have no problem constructing O(n^100) or similarly-ridiculous problems. Here is an informal survey [1] of several good examples. In my previous message, I indicated that matrix multiplication and CFG parsing are good examples; they are both cubic-time but we have come up with even cheaper algorithms for special cases, like sparse matrices or LR(k) grammars.
Whether P equals NP is not only a fundamental question, but it's the first of infinitely-many questions about the "polynomial hierarchy" or PH, a tower of ever-more-complex classes. And part of why the question is so important is that deciding whether P=NP is actually a problem in a higher rung of another complexity class further up in PH! P!=NP is like the first of an infinite family of very very difficult meta-proofs about complexity.
"RSA wouldn't break" because PRIMES is already known to be in P, via a famous paper titled "PRIMES is in P", and RSA keys are already sized to have impractical exponents.
What P=NP implies is much more powerful. First, let's consider the actual cost of a poly-time reduction. P is self-low, which means that P with a P oracle is equivalent to P. So if P=NP, then the cost of a poly-time reduction can be rolled in, and the entire problem is still in P.
Now, which problems would be transformable via P=NP? Well, we typically assume that the solution would include a poly-time reduction for 3SAT instances into some P-complete problem. Okay, great. However, we know that 3SAT can't be transformed opaquely; if this transformation exists, it must be able to examine and specialize the structure within each specific 3SAT instance. So we'd get one powerful theorem about logic (specifically, about SAT). But that wouldn't be the end.
We'd also get transformations for the following NP-complete problems, and each transformation would embody a free theorem about the nature of the structure of all instances of the problem ([2] for a bigger list):
* Graph theory: CLIQUE, HAMILTONIAN, K-COLORING (faster compilers!), COVER
* Number theory: TSP (faster packet routing!), KNAPSACK, SUBSET-SUM, PARTITION
* Formal systems: BOUNDED-POST-CORRESPONDENCE (holy shit, bounded Turing-complete analysis in P!?)
That's some serious free theorems! There's no reason to expect that whatever would give us so much stuff for free would be easy to prove. In fact, it's quite the opposite: The fact that any NP-complete problem, if it yielded, would give us deep insight into all of the others, should give us extreme pause before hoping that P=NP.
[0] https://www.scottaaronson.com/papers/pnp.pdf
[1] https://cs.stackexchange.com/questions/87073/what-are-the-ex...
[2] https://en.wikipedia.org/wiki/List_of_NP-complete_problems
