A single study can't get close to 100% certainty, because that's just not how science works. ... the fact that scientific methods are designed under the assumption that studies will be repeated, etc.

The current scientific method is in no way ideal. If a study were properly Bayesian, then you should be able to confidently learn from its results. That still leaves issues of trust and the possibility of human error, but there might also be ways to combat those. But in a human society, repeating studies is perhaps the best thing one can hope for.

The steps within unsound mathematical proofs may be valuable, but their conclusions are not.

Agreed. That is the one part of an unsound proof that is useless.