I think it's wrong to think of Science/Nature as aiming for a higher technical standard (or, at least, achieving it). They claim to distinguish themselves by two qualities, broad general interest and speed of publication. Speed is pretty much antithetical to highest possible technical review. One of the problems with the current situation, as you point out, is that the community has accepted "published in Science/Nature" as equivalent to "higher technical quality than the next paper", which is not (necessarily) the case.

I do understand that what you're talking about is substituting journal-driven peer review for community-driven peer review. My point is that peer review is real work, that substitutes for work you could be doing on your own papers, and it's hard enough to get good peer review when you apply a good deal of skill and effort to obtaining it.

You express a hope that the kind of community-based commenting and review that happens in mathematics will be eventually seen as valuable. This would be great. It would help if such community service were seen as valuable by search committees and promotion committees. Could we create a website where "the best public peer review" gets aggregated and credited? Then you could put it on your CV and get some benefit out of it.