A (Probably Terrible) Idea to Help SCH class identity

[YusiKha]

The last AST I had...

Well yeah, all of this talk is in the context of 2 competent cohealers. You can't judge job design from how the worst people use them, because there are thousands of ways to play jobs badly, and only 1 way to play them well.

I imagine it's actually bad to try and help your AST with a share of the healing when it's available, since they want to take center stage and use it and you'd just be causing overheal.

Correct. I always call out 'star's down' to my cohealer to ensure that she doesn't waste resources healing when I have it covered.
Assize wants to be used on cooldown too, but doesn't have an automatic 10s warning for your cohealer, nor a super high healing potency (400 is nice, but Shadowbringers' formula change means it isn't as obvious as it was before).

The game of trying to minimize dps loss...

Now this is the part of healing as WHM or SCH, Heck WHM + SCH especially, that I dislike so much. It's not a game of managing your DPS tools to freely heal, but a constant DPS loss that someone has to bite the bullet for.
WHM has no free healing outside of Dia refreshes (of which Assize takes a weave slot in half of), since Afflatus is a built-in damage loss, but Asylum, Tetra, and Bene are all immediately strong and useful.
It blows my mind that it's optimal to just ignore a job's main mechanic

SCH has access to free weave slots (3 per minute plus Dissipation, which is also your movement), but the only skills that can be freely used are Embrace/Seraphic Veil (Unconrollable 105p/280p) Whispering Dawn (580p), Fey Blessing (245p), and Fey Union (280p/tick). Compare that to AST's free everything
'Oh but Yusi you also have Seraph's Consolation and Recitation too' Those both require 2 energy drains to use! 3 for both Consolation stacks! A 2 minute cooldown and all 3 aetherflow stacks used on uncontrollable single-target heals and 840p worth of healing and shielding.