Personally, I use toothbrushes for a variety of cleaning purposes, only one of which is brushing my teeth (although I do prefer not to use the same toothbrush for my teeth as I use for other things...).
OK, let me take a different approach to this issue.
My PC is being chased by a pack of wolves. He casts Phantom Steed to escape from them. A phantom steed has a speed of 100 feet, so it quickly outpaces the wolves. The spell has a duration of 1 hour. Riding a horse is light activity that you can undertake while having a short rest. At the end of the spell, I've had a short rest and, as a result of having that rest, can spend Hit Dice to regain lost hit points.
Is Phantom Steed a healing spell?
See, the bit that I really don't understand about this discussion is that, if anything is providing healing to a character, it's the short rest, not the spell. There's a logical leap being made between the casting of the spell and the healing that just doesn't exist. It's like saying Charm Person is a divination because the creature you cast it on is friendly to you and will answer your questions.
My point is not semantic. It's the point I've been making for a while now. Do not confuse the effect of a spell with an outcome that the spell might achieve. Even if that outcome is commonplace. It is the effect that defines the spell, not some downstream outcome that is contingent on the circumstances in which the spell is cast and its application in a particular case.
Cheers, Al'Kelhar