You're thinking in terms of the traditional peer review model. I'm imagining a more organic process where papers are scrutinized to the extent that they're considered important or promising. For example, how Grigori Perelman's papers on arXiv about the Poincaré conjecture were reviewed by peers. There's no assignment of individual reviewers in this model, it's a natural byproduct of researchers discussing and collaborating in public.

Grigori Perelman's papers were considered promising mostly because they were written by Perelman. When there are too many papers to read, the reputation of the authors becomes the most important signal of quality. Formal peer review tries to spread the attention a bit more evenly, but it doesn't scale too well.

On the one hand yes, Perelman was known within his subfield. On the other hand, I find it hard to believe that if some random person had published the same papers, they would have gone unnoticed, since as far as I understand it these were important results that would have been worth evaluating based on the titles and the abstracts.

Volume isn't really a problem going forward, at least in mathematics, since every proof can be accompanied by a Lean verification. Whether other disciplines can implement automated verification or quality assessment will depend on the details of the field.

The more famous the problem, the more reputation matters. There is no shortage of papers claiming to prove P vs. NP in either direction. Nobody is taking them seriously, as everyone has a strong prior that people trying to publish such results are cranks.

Mathematicians need to rethink what is the purpose and value of doing pure mathematical research in the future. Until know, they have mostly valued the creation and systematization of human knowledge. From that perspective, Lean proofs mostly increase confidence in the human knowledge. But they don't contribute directly to it, as a formal proof is unlikely to contain insights that are not present in the human proof.