I’ve had the exact same issue after installing the Developer Beta iOS 27.0 (24A5380h); I downloaded the Developer App and followed this method; yes both the Public and Developer options appeared in the update setting… but when I selected Public Beta to install beta 4 both options immediately disappeared again and the ‘Checking for updates‘ is still looping indefinitely,
I’ve had the exact same issue after installing the Developer Beta iOS 27.0 (24A5380h); I downloaded the Developer App and followed this method; yes both the Public and Developer options appeared in the update setting… but when I selected Public Beta to install beta 4 both options immediately disappeared again and the ‘Checking for updates‘ is still looping indefinitely,