As another example, consider the question of whether there exists a right-angled triangle with rational sides, having an area of 157.
Once he had 1e-33 for all n > 100 that could mean that even trying with 1M computers where each tries 10G solutions per second (1e6*1e10) some millions of years could pass without the positive result. Then it's exactly reasonable to say "for my money Fermat’s theorem is true" as in, really not worth trying blindly.
Imagine somebody came to you at these "early" times (Wiles proved the theorem in 1995) with a "grand project" to use a lot of computers to search for a possible "solution," being able to try 10 billion Ns in one second. What would be your argument against such a project? Would you try some similar derivation to get an estimate of success?
I know, brute forcing all integers is impossible, but you can even "imagine" a "superquantum (and now not existing) computer" which can do a lot of small integers in parallel.
And yes, I know, probabilities aren't proofs, especially not for integers. One counterexample is enough:
3987^{12} + 4365^{12} = 4472^{12}
Also interesting to see the accidental(?) order of magnitude of the numbers involved.