... so replace all of SCH's filler skills (which of the list you described only actually includes Bio and Energy Drain) with abilities that have the same effects, but new names and visuals.
Okay. Given they already changed SCH's Bio to "Biolysis", all it needs is a rename to Energy Drain (or for SMN to have Energy Drain renamed), which already doesn't function the same way SMN's does.
What I find quite interesting is that you specify "all skills that belong [...] basically to a Green Mage" -- particularly since none of the skills you named have ever been in a Green Mage repertoire.
Maybe I'm splitting hairs a bit since different editions of FF12 consider Poison or Drain "Green Magick", but Green Mage isn't officially a class there and those skills get shuffled into Black Magic between editions. In any game where Green Mage exists as a full job (which is... one game), Green Magic is exclusively either damage-mitigating buffs or crowd control effects (which all existing healers have), without even the means to deal or heal damage.
You have to remember that in classical FF games, SCH was sort of a Red Mage with support abilities like Scan in place of any Physical skills -- able to use both White and Black Magic. In that regard, Scholar has a better claim to Bio or (Energy) Drain than Green Mage, having had access to such spells for longer, more consistently, and in more games overall.
But let's discuss that. I'm curious why you want to remove everything from SCH that resembles a Green Mage, then specify giving Scholar "its own mechanics" when the only mechanics it still "shares" with SMN are pets and Aetherflow, both of which it uses in a completely different manner. Particularly interesting is that you put the onus on SCH to change purely because some elements of it still resemble SMN, when SMN has already been moving in a new direction!
To be clear I'm not saying SCH lacks problems, I just think you're misattributing them.
Besides, being "unique" in the manner you describe is only aesthetic, and wouldn't impact any of the balance impasses faced by healers.