You could go over a wired network.

You want a switch, not a splitter.

A splitter is for running two 100 Mbps connections over a single 4 pair cable, and it uses splitters at both ends of the cable.

That is correct, yes.

You could setup a steam cache/LAN cache server so you only have to download games and updates once though.

You’d want iSCSI, but everyone gets their own volume to avoid issues like that.