I can't add other people in HomeKit even though they have accepted the invite, they have the same screen but no controllability

Some users are experiencing this due to the Home Architecture upgrade that was offered during a previous software update. This upgrade was removed and will return soon. It’s unknown when. This has resulted in some invited users losing access to the home. See: Upgrade to the new Home architecture - Apple Support

See if these support documents have steps that they may have missed:

• Share control of your home - Apple Support
• Allow others to control accessories in your home - Apple Support