So, like Resto Druid or Disc Priest, the reason it'd be 'an advantage' is the gameplay feel. Setting up your HOTs on Resto for the big raidwide damage HPS check thing, then hitting that Flourish to double their output and watching the HPS bars just shoot up (especially with how HOTs tick way more often in WOW anyway) is satisfying to see. Likewise, other people might find the instant jump from 10% to 70% because of a well timed Power Word: Life on Holy Priest. The reason I think 'do damage causes heal' would be 'an advantage' is because, atm, we have a very large disconnect between our healing and our damage, when it comes to GCDs. It's jarring to swap from our damage kit to our 'pump HPS for JWaves' kit. With a proper 'do damage, causes healing' kit, the two would be much more intertwined, and feel a lot more fluid.
Vampiric Embrace is interesting-ish, but not quite what I'm thinking of. Like, I was playing Guardian Druid in DF, and we now have a talent called 'After the Wildfire'. Every time I spend 200 Rage, I explode a burst of healing (about 60k in my gear) to all around me. Which is nice, but that is what I'd call semi-autonomous. I can't control when it goes off, it's happening at times pretty much out of my control, and I'd have to purposely play bad to get it to be timed right (not using Ironfur just to get it timed right). No, maybe I worded it wrong and that's MB. When I said 'semi-autonomously' re: SGE, I meant more like 'when you're playing correctly, it's happening alongside the damage'. You're still having to do Disc/Resto style 'set up' windows before the damage comes in. There's still 'choice' about what to use, and when. But when you are 'optimized', it's something you're not thinking about per se, as you're doing it instinctively. And since the effects I went for in my suggestion last only 4 hits (but can be used as much as you want, as long as you have the MP to fund it), VE's not exactly the most accurate comparison. It's more akin to setting up Atonements on Disc, I guess. But instead of lasting X seconds, I made them last X attacks.
And yeh, I guess theoretically, the SGE doing their healing via damage isn't covering everything in the raid, just as they don't now. They'd still have to rely on the cohealer for some things. As an example, let's take a raidwide, idk Hemitheos' Dark IV in P6S. Big damage, no bleed. As it stands, one Ixochole and one PI Rapture, and it's sorted. So, what if instead, we had the SGE able to PanKardia/Krasis/Soteria to cause some healing when they do damage? This way, we instead can go Ixochole, 'maybe' a Rapture (the SGE damage-heal might cover it if there's enough augments used) and therefore the WHM can keep their PI for something else. Or we could go Asylum/SGE damage-healing, and the Kardia heals are all buffed by the Asylum, meaning the burst healing of Ixochole and PI Rapture are still available for something else (in prog, very handy). The 'reward' is that your combat flow isn't going to get interrupted by having to press Eukrasian Prognosis, if you're skilled enough to cover it via the 'damage causes healing, which causes shielding' augment (I forgot which one I assigned that to, maybe it's Zoe)