I'd consider more: SMN, RDM, DNC, BRD supports or loosely fits a more supportive classes sometimes, Tanks/Healers are generally also can be considered "support Jobs" as they're not primarily DPS jobs but do DPS still, I get some jobs like Melees give out raid buffs but I don't think just Damage buffs make a actual support Job. It's kind of hard to balance Damage Vs Support, Because Damage is normally the most important thing. I think the Issue is theirs no real middle ground the closest thing you'll get to a good middle ground is Tanks because you have some responsibilities of a Healer with mitigation, sustain and party raid wides but you generally have a middle of the road dps rotation... I think healer goes too far into the support direction right now a lot of your healing support and gcd actions are generally more of something you don't want to do, so being a healer is cycling your CD's similar to a tank with more focus on party then self, but a lot of healer is just spamming one damage button. With "support or utility jobs" generally what I think is the issue is that giving up a big amount of "damage" for utilities that the Healer or even tanks have, Now it can come into use having that extra utility but it's something that's a nice bonus at best, if we had "Support Jobs" ontop of tanks/healers we'd really need to redesign the game from the ground up.
Generally what I'd like to see is:
Tanks, should have more stronger and Impactful raid wides, I feel like they've took aggro management from tanks and not given them too much inplace of it, I think tanks having a bit more stronger and likely a extra raid wide between each other or something and some more importance with what they can do to help the Healers, I think tanks can play this sort of "support role" with a good engaging DPS rotation with a few more changes, I want to see generally a bit more responsibility on tanks. Right now I think the EW short CD's feel really good and strong, but in general I'd like some more group responsibility with tanks.
Healers, should have more to do, they generally have nothing fun to do outside of "support the party" one button damage rotations and dots are generally really boring to a lot of people, Astro is my favourite healer because while I still have one damage button and a dot, I generally have to think more outside of healing support. Not every healer should be Astro though, some healers like white mage should have more interesting things inside it's rotation such as procs, I want it to feel like a Caster and a healer personally (doesn't need to have anything too overwhelming).