Cannot share control of home/homekit devices

I have the same question that it looks like a number of other users have had. Here is the summary of the issue from another user which is exactly my same issue:

CANNOT INVITE PERSON TO JOIN HOME APP

When I invite a member of my family to join Home App, the person I am inviting cannot join.

After a short while my user (apple ID) disappears and the user of the person I have invited shows as the Home Owner.

This happens also if I reverse the process, using the other family member's ID to create a new HOME and being invited to join.

Tried to log off from iCloud from all devices and redo the entire process - same problem.

Read and tried a lot of suggestions from different forums but still the same problem.

Anyone encountered the same problem?

Has anyone found the solution for this problem? Please note that I've already followed the standard Apple advice (i.e., both devices are on the same network, both devices have been updated to the latest iOS software, etc.). I would note that Apple suggests that you check to see whether both devices have two-factor authentication but -- critically -- Apple never says whether this is a good thing or a bad thing. Should we NOT have 2FA or should we keep it?