We're on Comcast Business class, but I still went with my own modem, picked up a 6121 for <$40, my rental fee was $12/month, so that was quickly offset (and I've had zero technical issues).
That router is probably _ok_ (not all routers are created equal, from specs, to "real" throughput, features, etc.) Your suggested setup is exactly what you'll want to do:
coax -> cable_modem -> router -> wired_wireless_connectivity
If you want/need ethernet (wired) connections to a few machines/devices, you could do those cable runs yourself, it's pretty simple.

Alternately, setup a bridge (a device/router that connects via Wifi, then supplies local wired connections).
We've got a router, a couple of switches (8-port on the AV system with 6 ports being used), a range extender here at the homestead