How do I remove a push-pull tub drain stopper?
I had a drain in my tub very much like yours, never found a way to remove the stopper.  So I can't answer that portion of your question.

However, to remove the gunk/hair that continually clogged the drain, I found the plastic drain sticks to work really well.  I just snaked one around for a few minutes, pulling up some really nasty clumps of hair and other detritus.

The "good news" is that now that the tub is 20 years old, the mechanism rusted out and I can finally remove the stopper.