I've only been to Rugby once, many years ago. (I seem to remember it as smaller than I'd anticipated and rather boring!). As I was driving, I didn't go into any of the pubs.
As you say, 'Sky Sports' can be a negative point with many pubs but my Sunday evening regular (which is a Wetherspoons place) has Sky Sports and most people hardly even notice that it's there. (It certainly doesn't adversely affect the atmosphere as the sound is usually turned down). I suspect that the Merchants Arms may be a sport-focussed pub during the evenings but it's more a restful real ale boozer (with decent grub) during the day. The only interior shot here makes it look like the type of place I might enjoy (and I certainly don't go to pubs where Sky Sports is blasting out commentary at full volume to loads of inebriated football fans!):
http://merchantsinn.co.uk/s_photo.asp
So your choice of venue might be determined partly by the time of day (or night) when you expect to be there. I note that the place you've suggested only opens during the evening during the week (and from 3pm on Saturdays) and operates a 'bookings only' policy in the restaurant:
http://www.quigleys-ontherocks.co.uk/