These are machines that have been managed for years. Restarting doesn't help. Total wipe may but is time consuming. Trying again in a couple of days may ... my current theory is that a wonky Apple server is contacted and it takes a couple of days for that relationship to expire so it talks to a different server.
Are there other workarounds beside web enrol? Not happy that associates a userid to the machine.
I think QuickAdd is very deprecated.
If you are on Monterey or Ventura and have either a Intel with T2 computer or an Apple Silicon computer, you can use the Erase All Contents and Settings to quickly wipe the computer (less than 5 minutes.) That probably is a better choice that having to Web Enroll.
Are you possibly turning on the computer with a network connection and then turning it off before enrolling? If you turn the computer on with a network connection and before it is assigned a PreStage enrollment, the computer will see that there is no enrollment setup. Even if turn the computer off, add it to a prestage, and turn it back on, it still wont update the enrollment token. You have to wipe the computer to get the updated enrollment information.
If you are already in the O/S, you can try running sudo profiles renew -type=enrollment to try doing a ADE enrollment from the O/S.