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/check/handle_array.cpp

@@ -5,7 +5,6 @@
 #include "toolchain/check/context.h"
 #include "toolchain/check/convert.h"
 #include "toolchain/parse/node_kind.h"
-#include "toolchain/sem_ir/inst.h"
 
 namespace Carbon::Check {