The pushed state is a replay, not a reading

A relay holding shared session state hands a new member whatever it last heard. That payload is correct — and its position field is frozen at the instant of the command that produced it. If the source has been playing uninterrupted for four minutes, the last command it sent was the item change four minutes ago, carrying position zero.

Apply it verbatim and the joiner starts at the beginning while everybody else is four minutes in. So the joining sequence has two steps, not one: