The subscription can lag. Phones decide for themselves how often to refresh a subscribed calendar. It usually catches up within a few hours, but iPhones can take most of a day. If you need a session in your calendar right now, use Add to Calendar on the session itself — that’s instant.
Add to Calendar is a snapshot. If you later cancel or move that session, the copy in your calendar stays where it was. The subscribed calendar is the one that corrects itself.
Keep your link private. It shows your own family’s sessions and names, and no one else’s — but anyone you send it to can see your session times.