BUG: DataFrame reductions with object dtype and axis=1
- [x] closes #49603 (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
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] 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.

In #49616 I attempted to only remove the errant cast to float, but this lead to the issue that, e.g. for axis=1 the sum of boolean columns becomes object. Unconditionally taking the block manager reduce path solves this problem, but then we run into the issue of taking a transpose of a frame with no columns (see the comments in the diff here). This PR resolves that issue, but is a bit of a hack (I can't seem to find any other resolution).

In the test here, there are some reductions (e.g. mean) that one may have expected float back instead - I've opened #49618 for this.
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. Closing in favor of #51335