I will use the following definitions
Platonic graph: A 3-connected planar graph with faces bounded by the same number of edges and vertices having the same number of incident edges.
(remark: the faces of a 3-connected planar graph are well-defined due to Whitney's theorem)
Combinatorially regular polyhedron: A polyhedron with Platonic vertex-edge graph.
Platonic solid: A combinatorially regular convex polyhedron with congruent faces of regular polygons.
Suppose we know the existence of the 5 Platonic graphs, but we don't know the existence of Platonic solids. How can we prove that they exist?
The existence of combinatorially regular convex polyhedra follows from the existence of the Platonic graphs by Steinitz's theorem.
But how can we know, that every combinatorially regular convex polyhedron can be continuosly deformed so that their faces become congruent regular polygons?