It depends on how the payment app works. Android provides a native Contactless Payments API which can be used by any wallet app. This is local to the device and works flawlessly on GrapheneOS as well. You can set your preferred wallet app for this feature under NFC settings.

Google Pay/Wallet is one of the wallet apps using this API. If you use Google Pay, you set it as your preferred wallet app, and Google will act as an intermediary between you and whatever payment method you've configured in Google Wallet. It's this Google Pay app that's broken.

Banking, payment and wallet apps that implement the Contactless Payments API work normally as they should. But, some banks have lazy developers, and just hyperlink you to add your card to Google Wallet instead.