A Live Activity is the iOS lock screen surface that tracks something happening right now: a timer, a delivery, a game score, updating in place without notifications piling up. Since iOS 26, Apple Wallet generates Live Activities for boarding passes, putting gate changes, delays and boarding times on the lock screen for the duration of a flight, fed by airline data.
The scope matters: in Wallet, Live Activities are a boarding pass feature only. Loyalty passes, membership passes, coupons and event tickets do not get one, and there is no key in the pass that turns it on. Apps can build their own Live Activities with Apple's ActivityKit framework, but that is an app capability, which is beside the point for a program whose value is precisely that it needs no app.
That does not leave loyalty passes off the lock screen. A push update that changes a field can carry a lock screen message, so "you just earned 120 points" lands the moment it happens. And pass relevance surfaces the pass itself on the lock screen at the right place and time, near a store or ahead of a booked session, one tap from the till.
So when a Live Activity comes up in a loyalty context, the honest answer is that it is the wrong tool by design: it exists for short, live events with a clear end, like a flight. A membership relationship is long-lived, and the lock screen tools built for passes, messages on update and relevance by place and time, fit that shape better.