As much as I love Cork for Saints and I think he is playing better than Cleverly I think he could find that level of competition too much to make a truly positive impact.
If we are talking World Cup finals then you may be right. However, many international games (i.e. the Denmark game) is probably a step down from the Premier league. Cork is never going to be the guy to come on and win you the game. What he is good at is keeping things ticking over, utilising other players and allowing them to do their thing. I think he could do that at pretty much any level and theoretically, he'd have better players around him finding space etc. If however, by "impact" you mean carving up defences, making THE telling pass, or scoring goals then, you are right, thats not going to happen.