This is entirely true, and it's the root of a lot of this. When I came over here from WoW and flipped to healer from BLM... I healed. In fact, I got annoyed at all these CNJ quests demanding I kill stuff because if I wanted to do that I'd be playing BLM (which is far, far better at it). Back when I was playing WoW, the endgame healing design was "you heal", especially if you were a MT healer. It was entirely possible for a tank to go from full to dead in three seconds from standard damage in a lot of fights and you simply had to have heals going out reliably.
It didn't click right away that the game design here is very much not that. That's the expectation a lot of players have when they come in. You see it all the time from new players, and other job players that start healing for the first time. I'd never kick someone for it outside of Ex or Savage because content below that often has people who are still learning, and that's not the way I prefer to handle someone who isn't actively being malicious. They're playing suboptimally, sure. But so was I, and people were nicer to me about it.
If you want to ask if the game should be that way, it's a complex issue that comes down to opinion on what you want the healing role to be. Changing it to be a pure healing focus would draw some people who don't like the DPS focused style, but would also chase off a lot of the healers we have now that do like it. It'd require major changes to how healing skills, incoming damage, encounter design, and even character stats are being done. Healing would be largely unrecognizable from what we have today if you did it.
For the game we have now? Look... the fundamental rule in any group is that everyone should be expected to do their best to help the group succeed. That means you use your kit to the best of your ability. Since each of us has different skill levels, that doesn't mean you need to have 99% uptime while doing a minimum of 2000 DPS as a healer. It does mean you have to try to help, and tossing up DoTs is more helpful than standing around doing nothing. Standing around doing nothing for 20 seconds (entirely doable on O2N as a healer after putting Regen up on the MT) can't in any way be construed as you helping the group. The examples from the other roles are me deciding to tank while doing zero DPS(*), which as a Paladin is entirely doable thanks to the miracle of flash spam. I'll just hit that button a bunch, then stand around doing nothing. The mobs are hitting me so I've done my role requirement, right? The main difference between the tanks and the healers is that absolutely no tank will defend that as acceptable.
*I don't actually do that. I do use Clemency a lot though, which according to some posters here is outside my job description as a tank. Maybe it is, but I prefer to not wipe, and Clemency is pretty helpful in that department.