Further refactoring of self update logic

[djc]

Previously:

• Simplify updates #4404

Followup from:

• rustup check: add exit status and no-self-update logic #4340

[rami3l]

Thanks for this patch! 🙏

The changes you made seemed innocent at first glance. However, the changes in the order of execution made me a little concerned, which is later confirmed by the test failures... 🤔

[djc]

Scoped this down to just the first commit -- I don't think I'm motivated to pursue the rest of these changes in the near future.

[rami3l]

I don't think I'm motivated to pursue the rest of these changes in the near future.

Many thanks all the same 🙏