It Shouldn’t Apply, Even if Just for Balance Reasons
Whenever you use a spell of 1st level or higher to restore hit points to a creature
Very interesting, as you are using a spell of 1st level or higher and your goal could be to restore hit points to a creature when you summon something like a unicorn. When the unicorn heals a target using its ability, it's only doing so because of the summoning spell you cast that brought it into being, theoretically meeting the requirements of this ability.
There are some pretty strong arguments against this interpretation—that a summon spell isn't a "healing spell" or that there is no transitive quality for something like this, that only spells which directly heal a target would apply for the ability.
"Healing spells" aren't some defined term, so a spell that has the end result of healing someone seems like it could be considered under that umbrella. Proving that there is some idea of transitive healing between the spell and the conjured creature healing someone would be harder, but it seems just as hard to conclusively say they wouldn't count.
One very important consideration is that this also opens up the single summon spell to trigger this ability multiple times with something like the dryad's 3/day goodberry, with each casting affecting up to 10 creatures (10 berries). So a single 4th-level conjure woodland beings with this interaction could heal something like 210 hit points! And at 7 HP per berry which each last 24-hours, this would even be useful in combat as a potion of healing substitute.
With those considerations in mind, I think rules-wise this would be fine to allow but that the balance issues shouldn't be ignored.