Here's my two cents on how to fix the fairy issues:

Fairy becomes an origin point for fairy abilities and no more. The actual casting is done by the SCH. The fairy will still cast Embrace or Seraphic Veil as she does currently. I think this should prevent the weird issues of pet abilities being wasted. Regardless of the fairy's Seraph state or not.

Going out of range of your fairy causes her to teleport to your location instead, in heel mode. I think this would be a nice quality of life change and there's little reason not to do this unless you just want to annoy the player.

Summon fairy becomes instant cast, but is still considered a GCD. I think some people might argue this is unfair to other healers, but SCH's main healing tools are oGCDs powered by Aetherflow, which is likely to be on cooldown and you lose all of them on death anyway. Not to mention the MP and fairy gauge. We have Dissipation to recover Aetherflow but to use it, we need the fairy to be summoned (since it sacrifices the fairy for 30 seconds). The point is having the fairy faster let SCH recover faster from a death, at least to state where we can pull off some emergency healing. Also remember that for GCD healing we have to weave Emergency Tactics.