A mathematician doesn't come up with a research where he says: "Ei look, we are now extremely confident that this theorem applies in every case... we don't have a full mathematical proof for it but I can show you all these cases where it holds true."
No, a mathematician proposes a result and then goes on to prove or disprove mathematically that it holds true under specific conditions in absolutely every case where those conditions are met.
Also they are not saying "all these cases where it holds true," they are saying "in all the cases, it holds true," which is the definition for validity.
If I have the theory, that all positive integers smaller than 5 are also smaller than 6, I would assume I'm mathematically allowed to 'brute force' the proof by enumerating all the numbers > 0 and < 5 and then testing for each that it's also < 6.
If I can prove I enumerated all the positive integers smaller than 5 and didn't find any exceptions, I guess I proved my theory...
It's a silly way to prove my silly theory, but it is a proof. (I have also a truly remarkable proof, which this comment is too small to contain. ;) )
Practically people will eventually say that if your test has been run N times using M different compilers and P different computers and they all give the same answer then you're probably correct, but having a proof that humans can reason through from beginning to end makes people feel more comfortable.
That being said computer assisted proofs are becoming more and more common and mathematicians are getting more and more comfortable with them as the verification tools surrounding them keep getting better. And even if computers aren't used for the final proof they are often used for intermediate tests. For example if you're making a statement about all primes it makes sense to quickly test the first few hundred million primes to make sure there aren't any obvious counter-examples.
It's kind of absurd the sorts of endeavors which computers have rendered not only possible, but relatively trivial.
> but having a proof that humans can reason through from beginning to end makes people feel more comfortable.
Saying that humans can't reason about computer-checked proofs is like saying humans can't reason about source code; which is to say, it's obviously false in the general case.
IAAMathematician and yes, it is still a proof.