if the only state is the connection itself at the transport layer
That's not really the case. Web socket connections exchange keep-alives as layer 7 payload. Others might argue that it's more accurately described as a sublayer somewhere between layers 6 and 7... but in any event, it is well-above the transport layer.
And many applications use web sockets in other ways that are also not stateless. Once connected, then authenticated, there's no need to continually re-authenticate, because the client on the socket now will be the same client 15 minutes from now, and this is overhead that would not be avoidable in a serverless environment -- every action on a websocket would need to be re-authenticated. For another example, with a constant data stream, the server might keep track of what has been sent or what specific subset of the stream the client is interested in.
If you aren't maintaining (or don't need) a persistent connection to a server, the question could be asked "why are you using a web socket?"
Perhaps also relevant: HAProxy, a commonly used load balancer with web socket support, maintains a persistent connection to a single back-end server for each current web socket connection. If that backend server goes offline, there's no provision in the balancer to choose another back-end for the existing connection. The client will need to reconnect.
