I'm right there with you all. I just "upgraded" from a 2020 SE to a 13 pro.
Even though when I set an alarm with a song that is on my phone (or, at least, worked with my 2020 phone), and the alarm app will play a preview of that song when I set it, all I get when the alarm goes off is the default sound (annoying!) and vibration.
Luckily, my Watch vibrates -- most of the time -- as well, so I haven't been late for anything... yet.
I am considering physically backing up my new phone on my old (2012) MacBook which has all of my songs on it... I will report back if this fixes the issue.
I do have Apple Music right now (giving that a second chance bc it was awful the first go), but if it is the source of the problem, I am going to quit that for good. I'd rather have my songs, ripped from my CDs (remember: 2012 MacBook, still going strong!) and purchased from my sources, than not be able to my play music for alarms.
I may put in a call to Apple after the holiday week -- IF I haven't returned this dumbphone by then. I wanted it for the good cameras -- but the current usability is making me H⁸ it. I won't even get into the throttled browsing power here.