Routines need notification permission. Aura asks the first time you create one.
If you declined, Aura will tell you and offer a link to Settings. Or go there yourself: iOS Settings, Notifications, Aura, and turn on Allow Notifications.
Check Focus modes too. If you have a Focus mode running, Aura may be filtered out even with notifications on. Add Aura to that Focus mode's allowed apps. This is worth checking specifically, since people using a screen time app often have Focus modes set up as well.
Aura sends: the routine reminders you asked for. It doesn't send marketing notifications.