DEPR: object inference in to_stata
- [ ] 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.

The "do X instead" won't actually work until 3.0, so this has to wait until then (and can't write the whatsnew until the 3.0 file exists). Putting this here so I don't forget about it.
This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. Now that we have the `3.0.0.rst` file now, interested in following up? Going to hold off until #55901 goes in it looks like we don't have any test cases that get to the relevant path.  editing test_big_dates to use `expected.astype(object).to_stata` fails pretty badly; it looks like in the delta=True case by storing microseconds instead of milliseconds. 

two options: first is to change the relevant path to call infer_objects() and then recurse, second is to just rip out the now-identified-as-buggy path.  preferences? Probably just take it out.  Simplicity is its own reward.
 Still interested in finishing this @jbrockmendel ? Thanks @jbrockmendel 