Will restarting Steam fix the "too many activation attempts" error?

No. The cooldown is enforced server-side by Steam, not on your local PC. Restarting the Steam client, restarting your computer, or reinstalling Steam will not shorten or bypass the 1-hour cooldown. The only fix is to wait the full 60 minutes without attempting any activations. If the error persists after a full hour, try switching to a different internet connection or activating via the web at store.steampowered.com/account/registerkey.