I don't know if clerical magic uses it or not.
As far as I know any magic that doesn't function in an anti-magic field uses the weave, this is one of the reasons deity magic still works. But you are saying that in 5e weave is not a medium but infact magic itself:
The weave is the fabric of magic which is woven into a pattern surrounding ferun.
What do the Gods use then? Or are they all subservient to Mystra since without the weave they would have no power?