Know exactly what the OP means, not sure how related my disinterest in the national team is to our club's success though - I know Liverpool, Everton and Spurs fans who all feel the same.
I remember going out to watch the qualifiers, making a night of it, screaming like a little girl when Beckham scored that free kick even though he was a rag.
Nowadays I don't even watch England. I find the team boring, our tactics boring, the attitude of the players and coaching staff boring, the predictability of the same old boys playing regardless of whether they are playing well (or even at all) for their domestic clubs,, boring...
Maybe I will watch the Euros and the WC itself. But even then chances are I would pick out an Argentina or Germany match, especially if we are playing Switzerland or Montenegro. No disrespect, I mean we are hardly better than either of those, but you know with England you are going to get the same old patterns and nothing to get excited about. I certainly think of myself as English, but that doesn't necessarily mean I want to watch a team whose core consists of Ashley Cole, Frank Lampard and Wazzock and have one idea only on how to play.
We don't even ever use a friendly to experiment, at least not beyond calling up the next young rag to blood them, surprised Zaha has not earned a call up yet (or has he?).