Trust me, you haven't flushed them enough. Hook up a junk shifter cable to it, clamp the end of the cable in a vise, and work it 100 or 200 times while periodically flushing it.
https://www.bikeforums.net/21152872-post14.html
The 600's are notorious for gumming up like this and equally notorious for taking this amount of effort to get them working again. It's just the way they are.
-Kurt