Simplify template part toolbar and align with reusable blocks

[jameskoster]

The latest work done to improve the reusable block UX yielded an update to the toolbar:

These updates can be ported to template parts as well to promote consistency across similar site building components:

For now I think it makes sense to restrict template part switching so that it is only possible to switch one template part for another. Transformations introduce further complexities and questions that I do not think we're quite ready to answer yet.

I begun work on this in #27397 and there is more context in the discussion there.

[mtias]

Also noting that the dropdown would need to account for patterns once we have them #27575

[Addison-Stavlo]

Noting we should probably have the renaming input that is currently in that toolbar moved somewhere accessible like the block inspector sidebar. Left a comment/idea on that issue for updating the block inspector - #27875 (comment)

[mtias]

Yes, it should align with reusable blocks there too.

[mtias]

With the decision that the "switching of template parts to another template part" is not a common path to warrant such a prominent space, this can be considered finished.