I think they really should have extended the Early Access period until they get these registration problems taken care of.

If everything was working as it should, then access would be cut off if you hadn't registered the game, and that would make sense. But everything is NOT working as it should, and they have a ridiculously high number of players in the same situation as you, with legitimately purchased games that can't be properly registered until their backlogged customer service team gets caught up on fixing these problems.