Because if you know who's supposed to be in quarantine you can take steps to verify that they actually are. And also because people are simply less likely to comply with rules when they know no one can tell if they're following them.
> I don't understand why you think an app that can promise to keep people's personal data private would gain lower adoption than one that was explicitly collecting such data. Maybe I've misunderstood what you're trying to say here but it would make more sense to me that an app that can preserve privacy would have higher adoption, higher trust and therefore higher compliance levels.
Again, because you're comparing "opt-in and private" with "opt-in and non-private". I'm comparing "opt-in and private" with "mandatory and shares user data with the health authorities" (which is different from "[opt-in|mandatory] and public" in that in the latter case any member of the public can get information about infections; I think South Korea are/were running a system with that level of sharing).