Actually, the Americans don't want to get paid. The was talk of that several cups ago, but it's never come up again.....and I think the Americans DO care and they DO wanna win.
One of the issues with the USA team is that they play one of these events each year. It's either The Ryder cup or The Presidents Cup and both are very demanding. I imagine it can be hard to get fired up for the event when you have to play one every year.
If you hadn't noticed, it wasn't until The Presidents Cup became a regular thing that the USA started losing The Ryder Cup.