There are two things that matter on a LAN: Number of packets, and total size of data.
When you broadcast your state from each client, you send the state of each client, to each client. The number of packets per frame per link is N (one from each client, plus the one upstream,) and across all links it is then N-squared, where N is the number of clients. When each client sends to the host, and the host broadcasts to all clients, the number of packets per link is 2, one upstream and one received, and the total number of packets across all links is 3*N, each client would send and receive 2 packets, and the server would receive N but send 1 packet.
The traditional client/server networking model would use about 4*N packets, but each client would only send and receive 2 packets, and the server would receive N incoming and send N outgoing packets.
Because each client needs the state of all other clients, you always get N-squared data growth no matter what, until you implement a smart server with visibility/importance filtering.
Btw, this is also why peer-to-peer topologies over the internet aren't all that efficient compared to client/server, because the packet count grows by n-squared, not just the total data size in this case, too.