This is definitely how I always saw it.

Q isn't arguing in good faith because he wants to see humans "rise above it" as it were. It isn't fair, but it is a quick way to reveal if their most base impulses will take over when under pressure.

Q sees human society in TNG as on the path to becoming better than they already are, but thinks they need some prodding to make it there.

It's almost Nietzchian in a way. He wants humanity to leave the worst aspects of our culture which are holding humanity behind, and that takes effort and self-reflection and consideration. Q is trying to kickstart that self-reflection, consideration, and effort. He knows Picard is a good man and can do it. He definitely has a "thing" for Picard and I think it is close to love. It is why he chooses him.

He wants Picard to no longer simply be a real mensch, but a real ubermensch.

Sisko doesn't realize its a test to make humanity better, and just sees a Nazi arguing in bad faith.

So he punched Q.