I'd like to add that I think that the "Everyone, Mature, Adults Only" distinction is a good idea that I've seen work on other sites, and that I think the middle ground of "mature" would be particularly nice for my current projects. The current two settings—"acceptable to absolutely everyone" or "so adult this game is hidden from normal browsing"—are a bit extreme, and I think there are a lot of works that hit a middle ground.
As for other types of content warnings… a lot of ideas have been thrown around, but I think simply lifting the 10 tag limit would satisfy the vast majority of use cases with a feature the site already has. I imagine the current limitation was instated largely to prevent tag spamming, but I think people here have presented use cases for more detailed metadata that are compelling enough that they may outweigh the moderation headache.
In addition, I think the ad hoc nature of tags is better suited for a job like this than something more organized and curated. I imagine that the types of content warnings people could potentially use are varied enough that attempting to make a set list of them would be an exercise in futility, particularly with adult content.
I do think there should be leeway for individual developer judgment with this, though. For example, while in terms of age ratings, Doki Doki Literature Club should likely be required to at least be marked "mature," I do not believe the developers of it should be subject to disciplinary action for saying "this game is not suitable for children or those who are easily disturbed" instead of something more specific. I'm not sure if this is even remotely in the cards, but it does seem like whether something like this could happen is very dependent on the details of how content warnings are implemented and moderated.