There are 3 basic kinds of wood overlays. One is wood itself. The outer layer is basically the same whether it is veneer (over solid) or plywood. The only thing to watch out for with this is sanding--going thru the outer layer.
Next is laminate. This is usually represented by brands like Melamine and Formica. These are water resistant and may or may not need sanding to paint. Since the plastic layer is thinner, sanding and not going through the layer is a hazard. There are some chemical products, basically available at specialty automotive paint stores, that you can spray on to make paint adhere.
Finally is the real thin plastic that is similar to Mylar. This is the stuff that balls up when you sand. Don't bother. It will be more expensive to do this reasonably than new--or restoreable solid wood pieces--will cost. Those last statements should be classified as opinion, I guess, but I think that they are pretty supportable, particularly over some time.