I heal, dps and tank in this way. It is by no means hard on the healer if the tank/dps know what they are doing. In fact chain pulling in a skilled group is incredibly rewarding. I mostly do this as a dps or a tank as my healer is sadly an imp, but on the occasions that I do get to play him with guildees I love it.
I do not pug often and usually when i do its a second dps we are in need of but there are times we pug a healer. We do not make their life hell. In fact, we often get complemented on how fun the run was. Recently recruited a new scoundrel healer in this way. If you mostly pug I can understand your concern more as in the few pugs I have run alone or just with other dps I have noticed a tremendous decrease in quality of pugged players since as late as the summer when I pugged more frequently. A lot of the few new full pugs have been downright awful with people doing the mistakes listed above by several posters. I don't know if I just did not notice this all before as I was less skilled than I am now or if it is just that ftp has tanked the skill level of the average player. Perhaps a combination of both. In any case I would advise you find a good guild to run with and I am sure you will see the virtues of fast pulling a fp.
That said dps and tank need to know what they are doing as i said: it bears repeating. They need to know the encounters and how to dps or tank them well, i.e. how to kill the group and hold aggro most efficiently for the least damage. If they do there is really no disadvantage to the healer unless they cannot manage their heals at all. In such cases, me and my guildees do slow down but with a skilled healer there is no need. That is to say it is not "catch up mode" unless one of two things happens: the dps and tank are unskilled or the healer is. Most often it will be the former as it takes a lot to mess up healing good dps and tanks in fps. But there are casual healers and learning healers who understandably do.
Also I must say it is irresponsible to leave accidental aggro messes to those that made the mess. 90% of the time they do not engage in the pull purposely desiring to do it alone as was suggested. It may just be that they do not pay complete attention or do not know the flashpoint or that they screw up. If they keep doing it, yes it is obnoxious but calm down and realize that flashpoints are casual content. Do not expect every player to play to a high level of skill: they do not need to and they are not committed to. As such it is just plain rude to leave them to die. Learn to respect other players' game. Not everyone is here to play their class optimally and beat the hardest bosses, some just want to taste the lowest hanging fruit of the eg and there is no problem with that. So when you are among them, be respectful and courteous.