Nope, no "router". Spectrum (Time Warner) gave me a Hitron Technologies E31N2V1 "modem". It also has the Spectrum name embossed on the side. The purpose of the Netgear GS608 switch is to replace any kind of wireless router. In effect, it serves as a "wired" router. I had a wireless router for awhile but it cut my Internet speed in half so I trashed it.
Everything I've gone through on this problem would indicate that you're probably right that the Spectrum modem only allows one IP address. I wonder how connecting the Roku first and then the computer would over-ride that problem? But, as I mentioned, I think if I turn off the Roku that it will lose its' Internet connection and won't be able to find it again unless I go through hooking each item up separate and the computer last.
Keep in mind that most wireless routers include LAN ports. You don’t have to use the wireless, just the LAN ports if you wish. I have a switch connected to one of my LAN ports, as I connect everything permanent via LAN cable, and use wireless for phone, tablet, and other portable devices.