Hi Martin,
Great question!
I am pretty sure, that it is actually impossible to do that.
Here’s a visual of what is happening when you Bevel Vertices in 2 cases:

A Vertex with n Edges connected to it, will Bevel into an n-gon (3 Edges > triangle, 4 Edges > square, and so forth).
You’d need to at least start with a spherical Object, that has Vertices with all the same amount of Edges and I think that that is not possible, but I will try and put together a mathematical proof (this has probably been done already, but I’ll see if I can do it as well).
EDIT:
It is definitely possible to have a spherical Object with all vertices having the same amount of Edges…the 2 bottom Spheres already have that…but it just isn’t enough.