I have a PhD in an esoteric field myself. There's literally <50 people worldwide who write/verify proofs in it. I've spotted errors in my own work or other's work that got past peer review. And I don't think I'm particularly special. There's just not enough eyeballs, time and incentives sometimes.
It only takes one to break it. And, for something so fundamental, it's a big deal if you are the one.
This isn't physics, where you have to rule out all manner of confounding data before you can trust your conclusion.
For example, given the axioms of Euclidean geometry, I can prove that the sum of the angles of a triangle add up to 180 degrees.
That statement is true. Full stop.
Now the starting axioms may not be correct (the parallel postulate, for example, is not required to be true). However, given the starting axioms, proofs are true. Period.
In addition, there will be things in mathematics that you cannot prove--Godel's Incompleteness Theorem sits in this section.
And, science has the problem that a hypothesis can only be disproven.
Mathematics does not suffer from the same issue, though.
I think you are understating the certainty of fundamental proofs like this.
Once such a fundamental proof is considered "true", it almost always has applications beyond what the original proof was used for. It is often used to try to prove things that we already know are "true" in other ways.
So, any "true" proof generally gets tested from multiple directions.
Wile's proof is a good example of this. It doesn't just prove Fermat's Last Theorem. Quoting Wikipedia: "Wiles' path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems."