Is the purpose of theoretical physics purely to seek mathematical innovations with no connection to objective reality?

Mediocre by what metric? American STEM education seems to objectively be doing pretty well in terms of Nobel prizes, scholarly journal articles, patents, technology product revenue, etc. Of course there's always room for improvement.

No, that's also misinformation. Public charter schools in most states aren't allowed to screen prospective students up front. Any parents can enroll their children, and when a charter school is oversubscribed they use admission lotteries. And they follow the same disciplinary procedures as other public schools.

A lot of the complaints here about physics have to do with focusing so heavily for decades on string theory (or M-theory) which hasn't produced much in the way of practical results. At some point we have to quit throwing good money after bad and redirect funding towards other lines of inquiry.