I’m having the same problem and haven’t been able to find a good solution either. I’ve signed up for several public betas over the years without any issues until I think iOS 14 when I discovered the same sign-in/welcome page loop you described. This year for iOS 15 my workaround was to download the profile under an alternate Apple ID I already had set up previously for an unrelated issue, but that’s certainly not ideal as you mentioned. Thankfully the profile downloaded under a separate Apple ID still installs fine on a device signed in to my primary account. However, I have yet to gain access to Feedback Assistant since my primary Apple ID isn’t enrolled in the beta. I’ve seen several others in multiple forums with this same issue but so far no actual solutions, and calling Apple never seems to have helped. Very confused.