Why can't I authorize my Mac mini on Apple Music despite deauthorizing other devices?

Your second screenshot shows "associated" devices which are different than "authorized" devices.

To deauthorize devices, see the information in Authorize or deauthorize your computer for Apple Account purchases - Apple Support

Note that if you have authorized devices that you no longer have access to, you will need to "deauthorize all" and then reauthorize any computers that you still have and want to use. Note that this only involves computers, not iPhones or iPads. See the following from Authorize or deauthorize your computer for Apple Account purchases - Apple Support

If you need to deauthorize a computer that you don’t have access to

To deauthorize a computer that you don’t have access to, you have to deauthorize all of your computers. Then authorize each computer that you have again. You can deauthorize all computers only once a year.

1. Open the Apple Music app or Apple TV app.
2. Choose Account > Account Settings.
3. Sign in to your Apple Account. Forgot the email address or phone number that you use with Apple Account?
4. On the Account Information page, look next to Computer Authorizations, then click Deauthorize All. If you don’t see this option, then this is your only authorized computer.
5. 
   
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7. Authorize each computer that you use again.

Also, for more information on authorized vs associated, see: Authorization vs. association - Apple Community