Why Are There Finitely Many Regular Polytope Compounds?

Question

I'm interested in a proof of the finitude of regular compound polytopes (vertex, edge and face-transitive unions of regular polytopes) in dimensions greater than $2$. To be clear, I don't need an enumeration result, that's more detailed and more obscure. I just want a clear intuition of why there would be finitely many, like proving there are finitely many regular star polytopes is very natural. However, the method I am used to there doesn't generalise, as the convex hull of a regular compound polytope need not be a regular polytope.