I had to Wikipedia "Cornhole" to find out what a cornhole set was.
Yes, if the Polyurethane is semi-gloss or gloss, then painting over
whatever paint you have with that polyurethane will give you much the
same thing as if you'd used a semi-gloss or gloss paint.
There are very many different kinds of urethane plastic, all of which
are called "polyurethane", but I presume you mean an "alkyd based
polyurethane" which is what replaced "varnish" as the clear coat of
choice over wood.
It's not a great idea to use a latex primer or paint here because the
subsequent coating with poly would be putting a hard coating over a soft
one, and that will result in the poly not being well supported and being
prone to "chipping" because the latex under it will break on impact
before the poly will. A better choice would be to prime with an alkyd
primer, cover with alkyd paint(s) and then apply your oil based
polyurethane to get the proper smoothness. That way, each coat is
strong and hard enough to provide good support to the one above it.
Note that if you're planning on leaving these corn boards outside, you
should paint the underside and edges of them as well to prevent them
from absorbing moisture from rain or the ground. If you don't do that,
the paint will crack and peel wherever the wood swells. Also,
polyurethane doesn't have very good UV resistance, so it would be best
to store these boards in the garage or basement when they're not in
use.
To promote the best possible adhesion of each coat to the next, your
best bet is to paint each oil based coating on within 24 hours of the
previous one being dry to the touch. All linseed oil based paints and
varnishes, all alkyd paints and all alkyd based polyurethane paints and
"varnishes" cure by the same chemical reaction, and so putting each coat
on before the previous one is fully cured promotes chemical crosslinking
between coats, and that ensures excellent adhesion of each coat to it's
neighbors. For a smoother surface, I'd paint with roller sleeves
instead of using a brush, and I'd store the roller sleeves tightly
wrapped in a plastic bag and in your fridge or freezer between coats.
The cold temperatures will prevent the paint or polyurethane from curing
while in storage.
And, if you use a flat or eggshell alkyd paint over your primer, I'd use
at least three coats of polyurethane over it to ensure the rougher
surface of the flatter paint doesn't cause any additional roughness in
the surface of the semi- or gloss polyurethane.
--
nestork