Unable to fix Apple Music login issues on macOS Sequoia 15.3

After updating to macOS Sequoia 15.3, Apple Music stopped working. Songs wouldn’t play and kept loading or showing a message about updating the music library. I restarted my Mac twice, but nothing changed, so I signed out of Apple Music. Now, I can't sign back in – after entering my email and password, the window disappears, and nothing happens. Thank you for your prompt responses :)

[Re-Titled by Moderator]