How to access old iTunes purchases associated with an inaccessible email on a new Mac Studio?

I recently got a new Mac Studio and, after loading my music onto it, I authorized the computer using my current iCloud account. However, when I play music on the Studio, I sometimes get a popup asking me to authorize a very old email account, a @roadrunner.com address that is now part of Spectrum. I no longer have access to this account and cannot authorize these songs because I don’t have the password.

I checked my iPhone, and the songs do not appear there either. Since I cannot enter the password for the old email, have I lost access to all my old purchases? If not, how can I retrieve them, and is there a way to identify which songs in my iTunes library are associated with that old email account?

[Re-Titled by Moderator]

Original Title: Authorized New Computer, but...