DEPR: Index.insert dtype-inference
- [x] closes #51363 (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.

Doesn't have a "do X to avoid this warning, not sure if this merits future option or temporary keyword.
Its kind of annoying that this warning gets issued (the PR currently suppress it) in `frame_or_series[new_key] = val`.  any preferences on whether to warn there? 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. @phofl thoughts on doing this?  i think there was some vague similarity to some of what you were working on.  main open question here is whether to continue suppressing warnings on setitem-with-expansion thx @jbrockmendel  Glad to get movement here, but i was hoping for some discussion of whether suppressing the warning in setitem-with-expansion was the way to go I wouldn’t want to warn there, I don’t think that anyone cares about inference there and getting rid of it is a pain 