Including C headers inside of `extern "C"` breaks use from C++. Hoist
the includes of standard C headers above the block so we don't try
to mangle the stdlib.
I initially tried to scope this with a targeted change but it's too
hard to do correctly that way. This way, we avoid whack-a-mole.
Firefox is working around this in their e21461b7b8b39cc31ba53c47d4f6f310c673ff2f
commit.
Bug: https://bugzilla.mozilla.org/1953080
