I think this supposed secret desire is so secret that it is in fact non-existent.
The reason the US guys don't seem to want to play is because they get ripped in the press after any loss. The press can't accept the fact that the European team is very good and deserved to win. If there hadn't been The Incident at Brookline, I imagine the European press would have laid into their boys following that collapse as well, but fortunately for them, they had a rallying point instead.
Look at the difference between the US squad after the Presidents Cup down under (98, I think) and the US squad last year. After the former, people were wondering if the Presidents Cup would survive. No one after the last 2 seemed to think that.
And, I'm sorry, but taking a week every year to go play in an exhibition like the Ryder Cup or the Presidents Cup just isn't even remotely taxing for anyone. Any excuse of being tired or whatever is just that, an excuse. If we were winning, things would look a lot different.