The question is not which is more powerful. It is how many people make it to the end.
An installed app can notify whenever you like, with no email permission and no dependence on the wallet.
Orders, payments, bookings, points and whatever else, in a place that is yours end to end.
If you get it installed and used, you win a place on the phone that no link can give you.
Between scanning a QR code and having the card there is no app store, no password and no 80 MB download. That is where most people drop out.
Your own app is several thousand in development and permanent maintenance on two platforms. For one shop the sums almost never work.
An Apple or Google Wallet pass shows a lock-screen notice when a reward is ready. It covers the case that actually matters.
Build an app if you already have thousands of recurring customers and it will do much more than loyalty: orders, payments, bookings.
Stay with the no-app card if your goal is getting people back, and you want it working this week.
Create my free card