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I notice this as well, the transmission is VERY notchy. It is the furthest thing from fluid and I'm reluctant to shift aggressively (and that's completely discounting the fact that it has the notorious second gear synchro issue). I'd love it if this short throw shifter solved that problem! My dad's '06 Z06 feels smooth as butter and apparently has the same transmission?
Your dad's shifter feels different because it is different. Your dad's [2006] ZO6 does not have the same shifter box as the one of your car. The ZO6 shifter boxes of that period are bolted solid to the torque tube whereas the non-ZO6 shifter box is mounted with rubber "bushings." It is these bushings that contribute the most to the imprecise shifting that many are experiencing.
A lot of work has been done to understand - and fix - the sloppy-bushing problem. Mine is fixed through the use of a MGW shifter, Red Line synthetic fluid in the transmission, and new [custom molded] urethane bushings... feels like a bolt-action rifle now!