Music keeps asking me to Authorize this computer- is there a solution to playing previously purchased music from previous iTunes aliases?

AFAIK, and I have been using Apple products since 2006, any purchases you make, like those from iTunes (now Music) is completely tied to the Apple Account (Apple ID) that you purchased them with. They are not now, nor then, transferrable to another account.

Also, you never "owned" the music; the music's artist does. You only owned a license for using it.

The bottom line? If you no longer have access to the original account, you would have to purchase them again with the new account. Sorry, but I don't know any way to get around that.

Your post had me curious, so I did a bit of research to help me understand what Apple has provided over the years. It may help you as well.

• Early iTunes purchases (2003): Songs were sold as AAC audio wrapped in an M4P file. The "P" indicated Apple's DRM (copy protection).
• 2007 ("iTunes Plus"): Apple began selling DRM-free songs as M4A files. These are still AAC audio, just without DRM.
• Today (Apple Music and iTunes Store):
  • Purchased songs from the iTunes Store are typically M4A (AAC).
  • Songs from an Apple Music subscription are also AAC-based, but they have DRM while your subscription is active.

Your understanding of the issue is quite correct, though less so your understanding of how music licensing has worked (in the US) since the 1960s and 1970s.

Purchased (licensed) Apple Music has not been encrypted and has permitted burning some CD copies for a number of years, whether M4P or otherwise.

The Apple Music subscription service (rental) does include DRM.

I’m unaware of a time when then Apple ID password resets were not available, but it has long been possible to lack enough context to convince Apple to perform the Apple Account password reset. Re-purchase the song where available, and replace this copy with the newly-purchased copy.