std: Don't deadlock/panic on recursive prints

Previously a panic was generated for recursive prints due to a double-borrow of
a `RefCell`. This was solved by the second borrow's output being directed
towards the global stdout instead of the per-thread stdout (still experimental
functionality).

After this functionality was altered, however, recursive prints still deadlocked
due to the overridden `write_fmt` method which locked itself first and then
wrote all the data. This was fixed by removing the override of the `write_fmt`
method. This means that unlocked usage of `write!` on a `Stdout`/`Stderr` may be
slower due to acquiring more locks, but it's easy to make more performant with a
call to `.lock()`.

Closes #23781

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