I would like to buy an unglazed ceramic pot that isn't painted brown. Do
they make such pots? I figure they do but all the ones I have bought are
painted brown and "breathe" no better than a glazed pot.

You are partly right. Most of the so-called unglazed pots are actually
slip-glazed in a matte finish. If you don't like brown, keep looking. Sometimes
you can find them in tan or gray.
However, why do you want the pot to breathe when it is not supposed to? Good
bonsai pots, unglazed or not, are stoneware, which is fired high enough so the
clay fuses and is waterproof. An earthenware pot, which is porous, would not
last very long, especially for a hardy bonsai which is exposed to freezing.

Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)