I also have Kohler bath fixtures in my 3 year old remodel. Is this the life span of these things?
I was lost, lost, lost, like you, for months and months, despite much googling. Finally I found the answer.
Most lift and turn tub drains have a set screw. In fact, everything you read makes it sound like they ALL do. You can find the set screw if you look just underneath the stopper with a flashlight -- you have to get pretty friendly with your drain. Make absolutely sure you do not have a set screw hiding under that stopper before you proceed with what I did.
But some stoppers actually have no set screw. How do you get them off? Lift the stopper part way up (all the way up and it spins around freely; part way up and it will not turn, providing the "resistance" for you to screw against) and then turn it counter-clockwise. This will unscrew the stopper from the rod that connects it to the drain.
This worked like a charm on my clogged drain. It took a lot more work on the one that was simply broken (the stopper would not stay up). The locking pin in that one had fallen out, so the stopper would turn endlessly and never lock, which meant that I could never unscrew it. After much struggle, I managed to jam what was effectively an awl into the hole where the missing pin was supposed to go, causing it to lock in place sufficiently, and then unscrew the stopper from that rod.
Now I have to figure out how to replace that missing pin.