We have lots of utilities throughout our codebase and we're not very vigilant about indicating which ones are intended to be public and which are private. Some methods like dask.utils._deprecated are marked as private

but most aren't marked with a leading underscore and also aren't meant for public consumption.

This lack of public vs. private specification causes issues like #10112. I think we should move utilities are that are intended to be private into a private module (e.g. dask._utils) -- we'll also want to emit deprecation warnings appropriately.

Note that we do have some public utilities, so this wouldn't be a blanket "move everything into private modules". But I suspect most things in, for example, dask.utils_test and dask.compatibility can be moved over into private modules (xref #10114).

cc @jacobtomlinson for visibility

Also cc @fjetter @crusaderky @hendrikmakait as I think the same argument applies to dask/distributed