rustdoc: Get rid of `CURRENT_DEPTH` thread-local variable

[camelid]

It turns out CURRENT_DEPTH has been around as a thread-local since the beginning of the rustdoc HTML backend, so it's very hard to get rid of it. The issue is that it turns out so many places in rustdoc implicitly depend on it, so threading (haha, get it?) it through all the functions is very tedious.

cc https://rust-lang.zulipchat.com/#narrow/stream/266220-rustdoc/topic/what.20is.20CURRENT_DEPTH

[jyn514]

This is tedious, but I don't expect it to be particularly hard.

@rustbot label: -E-hard +E-medium