Push/Pull drain removal

Another fun post from Miss Forum Newbie.

My upstairs tub/shower combo has a drain installed that I've read as being called a push/pull drain or a "clicker" drain. We've been having problems with the water draining sooooooooooo slooooooooooowly to the point that at the end of a shower (less than 10 minutes) I'm standing in water up above my ankles. ICKY! I've already used Drain-o because I had previously figured it was a hair/soap scum build up, but to no avail.

I wanted to remove the drainpiece itself to try using a snake but I can't seem to get it out. I can get the metal draincap and the rubber washer off but the plastic tailpiece(?) that has a brass screw(?) down in it will not budge. I've tried using a flat head screwdriver but all it does is chip off pieces of the brass screw(?). The whole thing will pop up and down but it can't be unscrewed and I'm worried that it could have possibly become corridied somewhere and I won't be able to remove it. Even with the metal draincap and washer off the water still won't drain any better so I know it wasn't a matter of the clearance to allow water to drain.

I swear if I can get this drainpiece out I'll retire the drain hole to a primitive rubber plug and be done with it, no complaints from me.

Here's a picture of your drain with the stopper and another picture of just the stopper. Tell me exactly which of the pieces you have removed and which you still want to remove. It seems to me that once the stopper is removed, you could use a snake if you wanted.

I think we need a picture of her drain setup. I think it's more like a standard pop-up that what your pictures show.