I guess I don't see the problem...
(sorry, I'm not trying to spoil things for the sake of spoiling)

Look at it this way, if each ball (when there are two) has an equal
change of being black or white (or both white, or both black),
then it stands to reason that about
1/2 of the time I'll pull a while ball.
(unless we find out that they are both black and I keep drawing from the same bag )

If you add a guaranteed white ball, it's logical that those
odd increase. So, 2/3 does not sound unreasonable.

I'm not seeing a contradiction.