Ah, I read that sentence as suggesting that one followed on from the other.
I'm sure it would help keep the country together if there had always been a UK or British team and league, but as it is there's always been a rivalry between the home nations, I'd say if anything it's less of an issue now in football than it used to be, given Scotland so rarely qualify and we barely play each other.
We do when it comes to the Olympics and military conflicts over the last few hundred years.