The discussion of what it legally is, and if/how well it works are two different things.
It takes a commonly available cartridge. The cartridge is fired. That makes it a firearm. Which sub category it best fits in is up to the gov's eventual decision.
One thing is clear, while the cartridge is "fired", it is not fired in the conventional chamber, so it cannot deliver the same effect. Its NOT like the end of the arrow is the muzzle of a .357 pistol.
Note the careful phrasing of the ad copy, "the power of a 357 bullet..." Since it does carry the bullet, and it does strike the target, with some energy, the statement is factually accurate. However it is not the same as the energy of that bullet if fired from a regular gun.