Just to give feedback - although it is a little embarassing:
At the time at some point we found out that the issue (non-visible 2FA field) was because of a CSS clause hiding one of the profile fields. With the activation of the 2FA the order of fields moved and then the 2FA activation button was hidden in the profile, instead of the original field That took quite some time to figure out.
Thank you for your persistent help, though!