Don't set `s_update_triggered` if the updater process fails to be
created. This will allow users in that situation to try to start an
update later without having to restart Dolphin first.

Ignore any new update check requests if one is already in progress.

Yellow squiggly lines begone!
Done automatically on .cpp files through `run-clang-tidy`, with manual corrections to the mistakes.
If an import is directly used, but is technically unnecessary since it's recursively imported by something else, it is *not* removed.
The tool doesn't touch .h files, so I did some of them by hand while fixing errors due to old recursive imports.
Not everything is removed, but the cleanup should be substantial enough.
Because this done on Linux, code that isn't used on it is mostly untouched.
(Hopefully no open PR is depending on these imports...)

Delete old Updater.log file from GetExeDirectory().

The log was moved to GetUserPath(D_LOGS_IDX) in 5.0-14529, but users
looking to make a bug report might find the old one instead so let's
delete it to prevent any confusion.

Gets these functions out of the global namespace.

currently windows-only