Actually plywood does not seem that much more than MDF. So, I could
afford plywood. (What I cannot afford is real wood like cherry).

I considered plywood but it seemed... less solid... (and more prone to
giving users handling the hexes splinters) than MDF. I do understand
that MDF has moisture issues which is a big disadvantage (as there is
some small risk of liquid spillage for my intended use). But does
plywood break apart? Do I have to worry about the layers of the
plywood separating?

Unfortunately, I do not have the tools (yet) to tackle this. Nor the
expertise to get to 1/64 or better precision. And more importantly, I
have a time constraint that will not allow me to buy the tools, learn
to use them, and get the finished product done. Although I do like
your thought process of justifying new tools.
Will have to try that on my wife in the future.