Yeah, something like this. I've seen various takes on it, but generally BRD, DNC, and RDM make the Support list, with various other candidates (NIN, AST, and SMN are the next runners up, for example). MOST of these Jobs (RDM aside) are already balanced around this idea of bringing support leading to lower personal DPS. BRD and DNC completely rely on that, as does AST.
I think the general idea/pitch is that it fills the 4th party slot in 4 mans (being 3-6 Jobs, it would obviously have the representation to be worth that), and 1 or 2 slots in an 8 man party (1 a given, with the second being one of your choices for the 4th DPS flex slot like we have now). BRD has been "arguably Support" since ARR, if not 1.0. I think BRD could use Protect at one point, even, and used to use the Healer LB3. For all of FFXIV's life, it was treated as a de-facto Support by players. MCH when introduced had some support abilities with the Turrets (for the life of me I can't remember it now, but I remember it being a thing then), further cementing "Ranged Physical" as the Support role of the game.
And, as you say, it could open up a decent chunk of Job changes and gameplay styles, like SMN having various support Summons, like FF games tend to have between buffing/defensive/haste/etc Summons balancing out the offensive ones in a lot of iterations of the Job. Offhand, Tactics had 2 healing, a shield one, and something else. FF9 Eiko had Carby for all three barrier types (Reflect, Protect, Shell) and also Haste (changed based on what accessory she had equipped), etc.
It would also help out a lot of the players that really do yearn for that playstyle. I don't think it's coincidence that a lot of the players who want Healers to be Supports tend to main (or speak highly of) DNC and RDM. I think it's a design fault that SE doesn't decide to actually embrace that. RDM, for example, is the perfect vehicle for "DPS that throws support heals" due to its nature and lore, which is what a lot of people asking for Healers to become Supports are genuinely asking for. Granted, the encounter design would need to be changed so that HEALING ACTUALLY MATTERS, but if RDM had, for example, Vermedica (at a 1000 MP cost so they couldn't just outright replace a Healer as they'd OOM rather quickly), it would give RDM's the ability to stand in for Healers in a pinch or support healing in situations that actually demanded it.
...that does, of course, require encounter design to actually demand it.
Likewise, BRD and DNC getting stuff like combat Pelaton and enemy debuffs.
I dunno, I think it's something worth considering, at the very least.