That is definitely not true about state management - you can build very TCP-like systems on top of UDP (see QUIC), and the final adaptation you need is to figure out how to make that work with multicast (a non-trivial adaptation, but possible). Once you do, you have the foundations of a pub/sub system in the form of a one-to-many (or many-to-many if you are brave enough) network protocol, and now you basically only have to figure out how to do state recovery of new clients and clients who are behind, and that can basically just be a database. The system described in the OP is kind of a combination of a database and load balancer in order to do this over unicast connections.
You have to do this state management with a TCP-based system anyway - the use of TCP doesn't magically come with the full state of past messages.