http://www.youtube.com/watch?v=44CQbWDDd64&feature=fvst - Might not stick with the 'Western' theme they were trying to emphasis, but I dare you not to find yourself singing it later today after listening to it.
http://www.youtube.com/watch?v=kYorZEdoaPU- More popularly known as 'Sway' by Dean Martin. The mexican version probably fits the western 'feel' more but both versions are great.
Hard to pick truly fitting songs. The 'Vegas' band circa Rat Pack is what I thought they would be going towards, but the music tries harder to pick up on 'Country Western' tones in the game.
http://www.youtube.com/watch?v=mzeR1pJIZBA
It's difficult to separate Vegas from the the Italian mafioso type songs.
Here are a few more:
http://www.youtube.com/watch?v=JJKTt30kMeA
http://www.youtube.com/watch?v=aS6-b7CONDI
http://www.youtube.com/watch?v=wDUBoSta4eY
Consequently, I drew some of these songs from the Mafia II soundtrack...

Big yes. We've pretty much put together the radio station I was expecting to hear with Mr. New Vegas. Classic Vegas showtunes on one station for when your in and around the strip. Dry & dusty country tunes for when you're braving the wastes. Add the exisiting fantastic ambient sounds and the soundtrack would have been covered with gold.
And yes, if I could build a "custom soundtrack/Radio Station" on my PS3/New Vegas game I'd be all over that...