Review _public_ API surface of our Server classes

Our server classes (Scheduler, Nanny, Worker) are using many, many non-underscored methods that are treated by developers as if they were private because using them requires rather low level understanding of the internals.

It is entirely unclear for users what is safe to use and it is impossible for developers to assess where a deprecation cycle is warranted.

This can lead to misjudgments like https://github.com/dask/distributed/pull/8189 where the clearly public API (via the Client) was preserved but the semi-public API of the non-underscored scheduler method changed it's signature.

While this is a bit of legwork, we should settle this once and review our API surface and rename/deprecate methods we do not want anybody else to use.