Remove headers marked as unused by ClangD. (#3661)

This required adding a few headers that were found transitively before,
but not too many. This is sadly a fairly manual process of opening every
file in my IDE, but I think I got everything in `//common` and
`//toolchain`.

There are a few cases where technically we don't need `foo.h` to be
included into `foo.cpp`, but I've forced those to stay with a pragma.

I've tried to catch the places where we can cut deps in Bazel as well,
but not sure I got all of those.

I had been noticing these in other PRs and it seemed better to isolate
the change.

toolchain/parse/state.cpp

@@ -2,7 +2,7 @@
 // Exceptions. See /LICENSE for license information.
 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
 
-#include "toolchain/parse/state.h"
+#include "toolchain/parse/state.h"  // IWYU pragma: keep
 
 namespace Carbon::Parse {