"Now, I say that this appearance of the Holy Ghost in a bodily shape, wherein he was represented by that which is a substance and hath a subsistence of its own, doth manifest that he himself is a substance and hath a subsistence of his own; for if he be no such thing, but a mere influential effect of the power of God, we are not taught right apprehensions of him but mere mistakes by this appearance, for of such an accident there can be no substantial figure or resemblance made but what is monstrous." (John Owen, Pneumatologia)
If the Holy Spirit is not a person but a mere outworking of the power of God, representing that power by a dove (a thing of the substance "bird" and subsisting as each instance of a bird with dove properties) would have been very, very weird and out of accord with how God has chosen to reveal the created order to His image-bearers.
Imagine running on a treadmill and the energy you expended burning up calories suddenly becomes a hamster??!! LOL.