Is there an easy way to replace faces with vertices? This is technically called a dual: https://en.wikipedia.org/wiki/Dual_polyhedron A simple example would be Create>Polygons>Platonic Solids[Icosadohedron] and if you replaces the faces with vertices, it would give you a Create>Polygons>Platonic Solids[Dodecahedron]. This is because naturally they are duals of each other. If you place on inside the other, you can see how the vertices match up perfectly with the faces of the other.

But figuring out how to do this with more complex objects is a little more tricky. I realize I could select all faces, extrude[faces not together, scaleXYZ = 0], then merge vertices to create vertices at the center of all the faces, but I'm still left with the rest of the geometry that I don't want. I can't simply select all the other vertices and delete them unfortunately.

Is there an easy way to do this? If you want an example to practice on:
Create>Polygons>Platonic Solids[Icosadohedron]
Select all vertices and Edit Mesh>Chamfer Vertices[.3333]
Select all faces and Edit Mesh>Extrude[local scale XYZ = 0, don't keep faces together]

and try to get the dual of that object because that's similar to the geometry that I'm working with.