Office 2024 won't activate on my Mac

The license out of date message usually means Office is using a different licence from the one you bought, so I would check the account first, not reinstall again.

First, sign in at account.microsoft.com/services with the Microsoft account you used when you bought Office 2024, and check whether Office Home 2024 is listed there. If it is listed under a different account, that is the account I would use in Office.

Next, in Word on the Mac, sign out of any Microsoft account that is currently shown, restart the Mac, then sign in with the account that shows Office 2024 in Services and subscriptions.

If that account still shows an expired Microsoft 365 licence, I would run the License Removal Tool again, restart the Mac, then install Office from the Services and subscriptions page for that same account, rather than from an old installer or the App Store copy.

Also check that the key was redeemed at setup.office.com. A product key that was never redeemed will not activate by entering it in the app alone.

If Office 2024 is listed under the right account and it still says the licence is out of date, I would note the exact error message and the macOS version, because an old expired Microsoft 365 licence left on the Mac is the most common cause I have seen for this message.