Ok so after several days of debugging, I found what could be wrong. Instead of triggering the Live Activity from a Capacitor plugin I trigger it from a Custom Native iOS code. Difference between both is very thin.

In the iOS project, Capacitor plugin is a Pod dependency whereas Capacitor Native Code is written directly inside the iOS app. It's the only difference I see, but I didn't find clear statement in Apple doc specifying that Live Activities can't be triggered from a Pod dependency.