I'm not sure any wording helps, because the reason it strikes us so poorly is that it's such an implausible idea. It might help to make it more concrete, to sa

I'm not sure any wording helps, because the reason it strikes us so poorly is that it's such an implausible idea. It might help to make it more concrete, to say “assume that we've proven this for say n = 3, all sets of three horses have the same color. Then I can prove it’s true for n+1, because my set {Auris, Brunellus, Camper, Diego} of four horses is the union of two overlapping 3-sets, {Auris, Brunellus, Camper} and {Brunellus, Camper, Diego}. By the transitive property, Diego must have the same color as Auris and the result holds. But obviously this works not just for 3 going to 4, but also 4 going to 5... Any set of size n+1 is made up of two overlapping sets of size n.” Maybe also someone can then intuit that the wool is being pulled over their eyes in this step?