BUG: mixed-type mixed-timezone/awareness
- [x] closes #55693 (Replace xxxx with the GitHub issue number)
- [x] [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.
- [x] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

Surfaced while implementing #55564.

Two implementation notes:

1) The implementation here is pretty ugly because we have to work around an existing deprecation.  I chose to avoid changing the behavior in any of the cases that currently warn.  Once that deprecation is enforced, we'll be able to simplify this quite a bit.

2) This takes a stand on #55779 for checking once at the end instead of inside the loop.
Thanks @jbrockmendel 