To be honest, from what I can see from the videos, the gearstick vibrations look to be within tolerances in the first vid, however it is more worse in the second vid and it doesn't look normal. Our cars have a vibration damper on the stick, take this off and you'll see how bad it actually vibrates without it. Before you take the gearbox out, i'd check the linkages between the gearstick, going into the gearbox for signs of excessive play.
Should you take the gearbox off, check the selector fork, release bearing and ancillaries. If all looks ship shape, then only would I look at changing the flywheel. Its worth changing just the clutch kit though if you remove the gearbox, just for piece of mind.