From what I have heard on the POTUS channel, there was a concerted effort to demonstrate that the treaty did not interfere with US sovereign power at all. The key group that opposed this can be gleaned from the last two paragraphs. The right-wing, usually fundmentalist Christian, home-schoolers did not want to have any requirements on how they take care of or educate their disabled children, and basically threatened the GOP lawmakers that any support for this bill would risk a tea party primary challenge. I haven't found much support for that claim, but it sounds valid. Reagan was disabled. Dole was disabled. McCain was wounded and probably qualified for disability status. They couldn't get 5 more Republicans to vote for this? (edit) This is the closest thing I was able to find supporting the home-school backlash against this treaty. Link (I'm also not looking very hard).