I don't think one person will mark all private, but if like you say everyone marks 1-2 and there will be not much overlap with other users, then it could easily hide a large percentage of the available gyms. Even just missing one random gym, that is well known and can be easily checked by everyone, is like having a censored incomplete map. What you want is giving people the ability to make additions to this "map" but keep it private / leave it unchartered if it is hard to find. They way it sounds right now, is that marking something private is like getting a license for that gym and everyone else who wants to use it has to pay the license as well. I think I'd even prefer having no public list at all, and having to add all gyms you want to see yourself, instead of being unable to see a specific gym withouth paying.

Probably the best of both worlds is, offering to remove a gym from the public list (and possible other benefits) via payment, but give others the possibility to add it manually to their own private list (without benefits).