I think that the way you read the description for Unicorn Spirit can change your interpretation. Personally, when I read this I read the part saying "if you cast a spell using a spell slot that restores hit points to any creature inside or outside the aura, each creature of your choice in the aura also regains hit points equal to your druid level" as "if a spell you have casted using a spell slot restores hit points to a creature, each creature of your choice in the aura also regains hit points equal to your druid level."
But I think other people read it as "If you cast a spell that restores hit points using a spell slot, each creature of your choice in the aura also regains hit points equal to your druid level." There is no way to know which interpretation is the better one, as both are equally possible.
One interpretation looks at the line "that restores hit points" as limiting the effect to only occurring when a spell is casted that restores hit points, while the other sees it as saying that if you restore hit points through the use of a spell then the ability activates.
I am personally of the opinion that the better ruling is the interpretation I initially had, as there is a 2nd level spell called Healing Spirit that creates an intangible spirit that lets you heal creatures that enter the spirits space.
This means that even if a creature enters the spirits space, it does not automatically get healed, meaning that the spell can possibly never heal, just like goodberry, while other instantaneous spells automatically heal.
The condition for the ability to activate should be the healing, not the act of casting a spell.