When we did the clippy survey, most people not using clippy did so because it had too much breakage. On the other hand, folks new to rust tend to turn to clippy for some automated mentoring (well, when it works, which it luckily does more often than not).
So in the spirit of improving on boarding new users, I think a clippy that works on stable is priority number one. So stable should gate on clippy and we clippy maintainers are prepared to put in the work to make it ok.
Gating beta on clippy would be an easy way to ensure clippy will work on the next stable, but not strictly necessary. I don't see much beta usage anyway.
With nightly, we have a kind of special situation, because some use it for other reasons than clippy, so they are likely to keep a nightly. Giving them a choice between the newest nightly and the newest nightly with working clippy may be the second best solution if we cannot make it work, provided we can make the rustup UX palatable.
Of course, also gating nightly on clippy would make for the best UX for clippy users, so we may want to try that route first and back off if we find problems with it.