It's hard to turn down a free legal education these days (assuming you would otherwise be taking out loans). However, I recently graduated from UCLA, and while many people who wanted to stay in LA did very well in their hunt for big law jobs (assuming that's the goal), ones who went north or east did not fair as well. For better national opportunities, I would stick to one of the other 4 - however, it is becoming more and more common for firms to primarily hire from schools in the area to cut back on recruiting costs, so if you really have your heart set on being in one city (even though you say you don't), you should go to the closest school to that city (as for Columbia/NYU, I think it's a matter of preference - you can't go wrong with either).
Chicago is really known for their academia and apparently their curriculum even caters to that a bit, so if you don't want to live in Chicago I would knock that off since you aren't looking to teach. All of the schools will give you great clerking opportunities, though less so at UCLA than the others (unless you are at or near the top of your class).
As for the Westwood dig, please. Brentwood and Santa Monica are fabulous places to live, and you'll be laughing your ass off at everyone else when you're studying at the beach on a 70 degree day in January.