I'm not sure if you are understanding what a router is / does?
A router shares one internet connection between multiple devices.
It is, however, no use if all those devices are switched off ...
Whether the router is connected to a modem or set top box, makes no difference to that
- I do apologise if I am misunderstanding your posts, but what other device(s) are you wanting to connect, when you don't need your main PC running?