The closed state is the feature
When the seats are gone, the page says so plainly โ and while places remain it can show how many are left. For a normal session this is logistics. For an invite-only one it is the moment the design proves itself: the forwarded link, the well-meaning plus-one, the colleague of a colleague, all reach a page that has already stopped taking names. Nothing has to be policed after the fact because the count did it first.
Watch the list as the days run. Registered names arrive, cancellations are marked, and the statuses โ registered, attended, waitlisted, cancelled, no-show โ tell you the real shape of the room before the session starts.