My mom's work is fairly similar. 7 points and you get written up, 3 write ups and your fired. But the points do reset every quarter and I think you can reset write-ups as well. Planned stuff like vacations and appointments don't get you a point as long as you request them far enough in advance. So it's far less stringent. I really like it, I wish I could implement something here with my team but HR says I can't.
Your faincee's system sounds pretty barbaric and I assume it will transform over time. Otherwise they'll end up firing everybody that works there.
I do like that can't stay lake to make up work rule. Allowing stuff like that encourages employees to not be there the hours you actually want them there. I don't allow employees to make up time unless we are super short handed, which we rarely are.