Your idea is a good one. The main way they want to use that "local" age flag still sends a signal about the user to the server. Even if it’s coarse, that's tracking and privacy concern.
I believe repeating what's worked might be the better option. Other media solved this with standardized rating systems. The internet could add a similar content signal alongside this. A similar content-rating header (G, PG, PG-13, R, M) could achieve many of the same goals as an age flag or boolean filters.
The ratings describe the content, not the user. Filtering can stay local, without disclosing anything about the user. Current parental controls rely on blocklists or detection, which are unreliable. A standard rating signal would allow them to be simpler and more consistent. Operating systems and/or browsers still have to allow controls, but that could look more like locally selecting ratings than strict age checks.
I think more critics need to recognize that without workable alternatives, others will push for these bad solutions. The good news is this isn’t new, and other industries seem to have mostly avoided making these systems legal requirements.