[libc++] Make __atomic_base into an implementation detail of std::atomic (#115764)

The __atomic_base base class is only useful to conditionalize the
operations we provide inside std::atomic. It shouldn't be used directly
from other places in the library which can use std::atomic directly
instead.

Since we've granularized our includes, using std::atomic directly should
not make much of a difference compile-time wise.

This patch starts using std::atomic directly from other classes like
std::barrier and std::latch. Changing this shouldn't be an ABI break
since both classes have the same size and layout.

The benefits of this patch are isolating other parts of the code base
from implementation details of std::atomic and simplifying the mental
model for std::atomic's layers of implementation by making it clear that
__atomic_base is only an implementation detail of std::atomic.