If Chemist does get added it will be a magical based healer using mp, because it has to thanks to esuna/respose/lucid dreaming, thank you cross role you limited how healers can be going forward because any new healer has to cast esuna/respose(for that 1 single quest at lv80), swiftcast becomes pointless if no cast time, lucid dreaming is there for mp.
Cross Role Skills are not an issue they can simply change the animation to fit Chemist in the same way that the cross action Peloton has a different animation for DNC/MCH/BRD.

Now could Chemist work like that sure their cast times can be potion mixing but what do you use for raise equivalent?, how about ogcd immediate heal? shield? only damage nuke 'spell'? only dot 'spell'? only aoe 'spell'?
There isn't any reason why they can't have a 8s casting animation for their raise that shows them mixing potions or charging up a mix. Or they could have a instant raise with a CD if square wants to give them something different.

When you go to answer these questions Chemist starts getting more and more restrictive yet it is a goddamn Chemist mixing anything and everything is what they do to buff/heal/hurt. FFXIV current healer design feels too restrictive for Chemist and that is a shame personally.
The game already has solutions for this, as said above not all role actions share the same animation, there is no reason why raise couldn't function the same as casting a spell just with a mixing animation and their regular rotation doesn't have to involve mixing everything. It's not like AST has to use their cards for everything they do. Limitations are things that we put in our minds with a little creativity there are no limits for what can be done, even for BLU.