I signed a player on Game Day week and the AI cut one of my inactive players (who I would not have cut!) during the same spin. That's broken: https://die-hard-fans.myfootballnow.com/team/7

Not broken.. it is as designed. You can't exceed the roster limit (or cap) on game day. If you need to temporarily go over the 53-man you have to do it on a mid-week spin - that's why it exists. Otherwise you needed to have planned your cut for the game day spin along with your FA signing.

You _typically_ cannot sign players on game day unless that has changed.

You have always been able to sign players on game day, the only exception is that the waiver wire is not processed on a game sim.