On naming: AbstractPurePath doesn’t seem right, as the class doesn’t have any abstract methods. And to me, PureLocalPath indicates the presence (rather than absence) of the __fspath__() method! I quite like BasePurePath, LexicalPath, maybe LexicalPurePath. Thoughts?
_BasePurePath is like PurePath, but lacking __fspath__(), __bytes__() and as_uri()
_AbstractPath is like _BasePurePath, but adds abstract stat(), open() and iterdir() methods, plus all methods that can be derived (like is_dir(), read_text(), glob()). It will be extended by tarfile.TarPath. If that goes well, we can drop the underscore prefix
_BasePath is like _AbstractPath, but it uses os functions to implement stat(), open() and iterdir(). It exists mainly to make _AbstractPath testable.
_BasePath is like Path, but lacking some methods like expanduser(), hardlink_to(), etc.
I may omit _BasePath if it doesn’t prove useful enough.