Agreed. A 5-port gigabit switch can often be found for $20-$30 nowadays. Even if a residential router with more than 4 ports exists, it will be cheaper and more flexible to use any router plus a gigabit switch.
I have tried this before with my old wireless router. The problem I had was that machines could not "see" each other across the wireless router and the switch.
Could that be because they need to be on the same sub-net (I could not change the setup of the switch)?
A switch, unlike a router, does not change the subnet. Your difficulties may be related to the "uplink" ports and/or the need to use cross-over cables. You normally don't need to be concerned with these with modern switches anymore.