L&LD
Part of the Furniture
If my router covers my whole house from the living room, should I just leave the router there, and have the switch, with all the cables in another room where it's not a visible mess to my wife?
Like I've already stated, have 6 or more LAN runs to the optimal location for your router. (The number of runs for me would be 10 or more; I plan on having an 8 port router like the RT-AC88U, eventually).
What you want is each port to have it's own cable run. These runs should still all be in another part of the home where you can add more network equipment as needed (including more switches).
Depends how you have your router and/or switches setup with how much of a visible mess it will be.
I like a customer who built a pedestal for his router in the optimum location in his office. He hid everything but the antennae inside the pedestal (which was about 7' tall) along with a fake plant which camouflaged the antennae very well. Two 16 port switches and all the electrical cables and power supplies were also effectively hidden inside too.
I would not recommend to have the living room be the termination point for all the cable runs. But I would recommend that all the ports on the main router in that room be used, if you have the wired clients to do so.