As this map illustrates, at one point the United States actually cared about watershed borders. While the deal with the 49th parallel was resolved (as much as I'd like it if the US kept the Red River watershed in mind) not long after 1820, the western lands of the Louisiana purchase followed the Continental Divide until the Civil War.
Why are watershed borders good? Well, I don't like square states, and nor do a lot of other people. Look at Russia, especially Siberia, where there are few square borders to be found, and even down to the district level seem defined strongly by natural features. Now, I think to have something so meticulous as what Russia (Empire, Soviet, modern) did is asking for too much, but as the example of the Unorganized Territory/Nebraska Territory shows, the US can follow the boundaries and hopefully carve out states appropriately using the Continental Divide as their guide. Canada uses the Continental Divide as most of the boundary between Alberta and British Columbia, for instance.
Is there any way to get the people drawing the borders to follow the natural boundaries to some degree? I don't think it's bad if there's a mixture of straight lines and natural boundaries (i.e. Maine). And what might the effect be on the states and territories? If, say, Colorado does not have the Western Slope on the west of the Divide, does that get awarded to Utah? Does Wyoming ever become a state?