DEPR: Enforce deprecation of ArrayManager
- [ ] closes #xxxx (Replace xxxx with the GitHub issue number)
- [ ] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [ ] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

Thanks @Dr-Irv, @jbrockmendel, and @mroeschke. I think this is ready for another look. Two outstanding issues:

 - [The arrays method is used in 8 places in pandas.core](https://github.com/pandas-dev/pandas/pull/57118#discussion_r1468915971)
 - Removal of `internals.base`. This will involve moving a significant amount of code around, so I'd like to handle in a followup. Thanks @rhshadrach 