I actually play all three, but for the most part I stick with DPS. It is not that I don't like the responsibility, I just like to be able to tank down a person or mob very quickly.
As far as most people, I imagine that they don't do it because there is no real reward for the extra responsibility. The game I came from to this, there was an actual reward for it, though it wasn't considered a reward. In that game, there was only one tanking class and one healing class. There were "off healers" and "off tanks" but none which could heal or tank at end-game. Well, they could if they were willing to spend the money it took to allow them to twitch with 5 attacks per second, but that is a different matter.
Anyways, doing any kind of run in a dungeon was very expensive, no matter what class you were. However, the tanks would have extremely high repair bills and they would almost always have to buy healing pots, mana pots, and speciality pots (for increasing defense, health regen, etc.) The healers had a single skill which was almost always necessary. It was a constant heal to the entire party and also reduced all damage to everyone by 50%, so it was very powerful. However, the problem was it would cost 500 MP every 5 seconds, so even when spamming MP pots, you couldn't keep up with it. Because of this, they had to buy an item out of the item mall which would recover their MP to full after it reached 75%, and this of course was also very expensive.
The reason I went on to explain all that was to justify the system we had in place. On these runs, we had a system. All of the drops were materials to build armor/weapons, so everyone had something they needed, which often crossed over with another person. So, one person would be the "banker" and keep all of the mats until the end of the run. After the run was over, the banker would list all of the items that were collected over the run and then would hand them out. The order was tank gets first pick, then healer, then dps by level, highest to lowest. Usually everyone would get something out of it, but the tank and healer would always get the most, and it was done this way so they could pay their bills.
Now, I never found the bills to be all that expensive, but I rolled a healer just for the purpose of farming mats for my other characters. Knowing that no matter what the levels were I would get second pick, I could get pretty much any mat I wanted and also turn a profit by selling what I didn't need or whatever mat was selling for the most at the time. In that game, you had a reward for that extra responsibility you took on for keeping everyone alive.
This game, you run flashpoints for gear, and unless there is someone else of the same class as you, you are most likely going to get the thing you want. Even then though, it isn't expensive to run them unless you wipe a lot. I'll tank or heal if I want to, but because I don't have any real incentive to, I won't unless I need something for that character. On top of that everything you get in FPs is usually bind on pickup, I can't make a profit from it.