A core, in a well-designed sandwich panel, does not need to have high tensile or compressive strength values. The core's mission is simply to add a material that is as light as possible, which moves the resistant layers away from the neutral axis of the composite, which provides thickness without increasing weight. The only thing the core needs is to have good resistance to interlaminar shear stress (shear stress in the plane is not so important). If you use plywood as a core, you are making very poor use of the sandwich material. Also, as has already been mentioned before, the resin impregnation of the plywood is much less. With a plywood core, you are placing very resistant material where it is not needed, that is, you add strength, and a lot of weight, where it is not needed.
I think someone is not clear about the technical reasons and advantages of using sandwich laminate.