See, that feeling of control is why I play healer main with a tank offspec. I decide who lives, and who dies. I can even save the boss, and let everyone else die!
To be more serious, the reason I enjoy healing is because I find it more interactive and engaging than DPSing. Apart from tasks everyone needs to perform (avoid fire, handle mechanics, etc), as a DPS your only task is to dish out damage. Perhaps throw in some shroud or stun or interrupt or whatever. As a healer, on the other hand, you’ll need to keep the entire group alive - you pay attention not only to the mechanics, you also pay attention to the rest of the party. And ideally, you also squeeze in some damage too.
But if you want to relax a bit, you just stand at the back and heal people. In many cases, your DPS won’t make or break the encounter, so there’s plenty of opportunities to just… lay back. Can’t do that as DPS, nor as tank.
Personally, I just don’t enjoy playing DPS. It doesn’t give me the overview of the whole party and the enemies we’re fighting the way healing does. Making those health bars go up gives quite a bit of oomph.
Hi. I’m very much not a social person. Yet, I main healer, and do just fine. Every PuG needs a healer.
Were I a DPS, I’d be in a lot more trouble, there’s an abundance of those.
The first - and only - MMO I played healer in is WoW. I started as DPS main, had a two-expansion tank stint, quit, came back as healer, and gonna stick with it (with a tank offspec, because tanking is still fun). In any other MMO or cooperative game, I never played healer, unless I could play a tank+healer hybrid (like a tank/healer Templar in ESO). When I played roleplaying games, I very rarely was in a support role. When I played classes that could heal and play support, I usually played those aggressively.
Not really fitting the stereotypical support role, am I? 
I know a lot of healers who are in similar shoes. We don’t see ourselves as support. We see the tank and the healer as the core of the group, who control how the run goes, with DPS supporting us with damage.