That's what peer reviewers are supposed to do! ...of course they don't get paid either. I think via Retraction Watch that a few journals have recently started trying some kind of ombudsman thing that does what peer reviewers do except for pay, but they're still usually someone who's made a career in the same academic discipline and are probably going to miss a lot of the same kinds of things peer reviewers do. I think to really do it right you need somebody who is good on science and stats in general but not a specialist in your discipline and with no horses in your race.

And for a lot of these studies, the problems are baked into the study design, so there wouldn't be a lot you could do at first other than reject a bunch of papers (or incentivize people to do fraud instead.)

I would totally do it if someone would pay me for it, though, yes!