Why not just abandon the idea of national teams altogether, then?
But nationality isn't cut and dried, it's ambiguous in many cases. If I were anywhere near good enough for international sport I'd be qualified for more than one country and I have feelings for both. Granted there will be doubt over the level of commitment in cases like KP and Trott but if you are to allow for the vagaries of migration and whatnot then you have to accept those.
Symonds was born in England, Nash is a Queenslander, Caddick represented New Zealand at youth level. They're quirks but I don't feel those quirks detract from the ideal of national teams entirely. They all had claims on the countries they ended up playing for.