I think you're overstating how easy it is to discount for wealth in those other scenarios. At least when it comes to high school research experience, it has rarely been the case IME that a student got such a position directly because of their family, and certainly it is very rare that a student explicitly works in a parent's lab.
Yes, some private high schools feed to quality summer research programs, but A) those schools give out more aid than you'd think and B) a number of public magnet schools have similar success in placing students with labs. I am sure that this does still correlate with wealth to some degree, but I don't think it does so in a way that the admissions committee can easily "see".
Beyond that, I think the vast majority of admissions committees (including all of the ivies) have little interest in discounting anything. I watched a lot of students that to be honest were just average get into very good schools, purely on good (inflated) grades from a name brand high school. The kids from my summer research cohort also got into very good schools for the most part, but those students were actually some combination of insanely smart and insanely dedicated.
It's a very hard problem for sure, I don't feel confident enough to say strongly that a particular set of metrics is obviously good or obviously bad. But I don't think removing standardized testing on the whole is likely to help with discounting for wealth (and I also don't think they really want to).
FWIW I think SAT/ACT are bad indicators for the "elite" schools because they just saturate and stop having much value in splitting the applicant pool. Especially the math section, it becomes a matter moreso of making a careless mistake or two.