Does Firebrand Raider's text "units don't ready during the Awakening Phase" affect everyone?

Yes — it's global. Firebrand Raider's last line is unqualified ("Units don't ready during the Awaken Phase"), with no "friendly", "enemy", or "here" restriction, so while it's in play nobody's units ready in the Awakening Phase — not your opponent's and not yours, including Firebrand Raider itself.

Two things it doesn't do: it only blocks the Awaken Phase readying action (Rule 416.3.a), so spells and abilities that ready a unit still work normally; and "Other units enter ready" is unaffected, since units entering ready never get un-exhausted (FAQ #12981).

It's also a static restriction rather than a readying ability, so things like Mageseeker Warden don't interact with it at all.