Radius of gyration gets you to the average velocity of all molecules in the rotating flywheel, basically, as if they were traveling in a straight line instead of rotating, so you can make direct comparison to the rest of the car which is traveling in a straight line.
Consider a 10kg flywheel in a 1500kg car... From GG's example in first gear, the flywheel represents (10*25)/1500 = approximately 17% of the inertia the engine is accelerating.
"Schiefgehen wird, was schiefgehen kann" - das Murphygesetz
Looking up radius of gyration on Wiki can clarify what it means to help with the understanding. It's a great way to do a simplified explanation comparing flywheel to car weight. Why'd you post that flywheel mass makes little difference? It certainly does, especially in the lower gears.