We struggled to get users added to our Apple Home smart home. When trying to add my sons, they just would NOT receive the invitation/notifications.
All software was updated. All accessories were up to date.
After contacting Apple support twice and not getting it figured out, my teenage son had an idea.
Simply enter a space (empty character) behind the Apple ID email of the user you are inviting before you hit enter. That is what worked for us.
Let me know if this worked for you!