Would you do it purposely? Do you see a big value in clinch training for street defense?
It depends. Yes.
Most real fights start at close quarters, so a knowledge of the clinch is imperative. If you can control the clinch, you can control the fight. As far as weapons go, they are always a danger (especially up close), and you have to have a good tactile sense in order to keep track of the movement of the other guy. Ideally, you want to jam the weapon as the assailant is trying to draw it--obviously doesn't always work.
It's a range that you should be familiar with, because, as pknox stated, you will probably end up there. People taking a beating don't hang out in punching range, they disengage or try to grab you. Since you probably won't get a KO right off the bat unless you hit just right, the clinch is a very common place to find yourself in. If you'll notice, most fights that aren't ended in the first few blows end up in a standup wrestling match that is sometimes followed by one or both of the participants going to the ground.
One thing to remember is that most people are very ignorant of the clinch. In terms of layman's combat, punching range is the most comfortable (almost everybody can hit with their fists to a certain degree, and even if they don't do it well, they are still most comfortable there). Landing knees, elbows, and headbutts can end a fight in short order, as can shredding. Your most useful weapons are in the close quarters area, so make use of them.
While you wont be able to get as much power in the elbow as say a punch
Not sure I agree here. When you elbow, you can torque your whole body into the shot, you are hitting with a much harder surface than you would be when you punch, your target is much less important (you're not going to break your elbow on somebody's cheekbone), and you will probably get a cleaner shot in (if you're in range to elbow, it's hard to 'block' them.
It's basically a grappling fight, with neither person in an advantaged position.
Definitely incorrect. There are definitely advantageous positions in the clinch. A Thai neck tie is one, double underhooks is another. The former allows you to strike with a lot of control, but the latter can allow for a takedown if the situation does not permit you to knee the hell out of whomever you're fighting. Grabbing clothing can make a difference, but if you train in a jacket and jeans, you should have an understanding of how the situations changes--doesn't change much, as the positions of advantage are still the same.