Yes that’s absolutely what will happen. They likely have language in the contract you signed allowing them to do just that.
You are paying for access to the gym. They don’t have to provide you access via a card or a list or an app - they probably specify that they can refuse access for a variety of reasons, including “safety and privacy” or some shit they can shoehorn an app into. You don’t have a legal right to access a place via the mechanism you choose.