DAS-2067 - Migrate datatree io.py and common.py

[owenlittlejohns]

This PR migrates what I believe is the last actual source code out of the xarray/datatree_/datatree directory into xarray/core. This is another piece of #8572.

I made a couple of decisions that I'd love some feedback/extra eyes on:

• I moved the remaining functions from xarray/datatree_/datatree/io.py into their own module, rather than xarray/backends/api.py. It felt like these were a little separate, but I can definitely be convinced otherwise.
• I created a subclass for TreeAttrAccessMixin. I really just wanted to swap over to using AttrAccessMixin, but I ran into issues with __slots__. Looking at it, I think I would need to work on the DataTree, TreeNode and NamedNode classes. I can go down that path, but thought I'd put up this PR for discussion before beginning that effort.

• [x] Continues #8572 (Contributed to last subitem of "Merge in code for DataTree class.")
• ~~Tests added~~
• [x] User visible changes (including notable bug fixes) are documented in whats-new.rst
• ~~New functions/methods are listed in api.rst~~