Why can't I change my Apple ID profile picture?

You might want to try changing it in your Contacts app. Just go to your card at the very top, and update it there. That usually does the trick.

I had a similar problem where my profile pic reverted to my initials after I deleted a duplicate contact card. It turned out that the photo wasn’t syncing properly. I managed to fix it by logging into my Apple account on a desktop browser (not on mobile) and changing the profile picture there. After that, I rebooted my devices and everything updated just fine!

You can also change it directly from iCloud.com if the other options aren't working for you. I faced the same issue, and that did the job for me.