I think the analogy works well but you must understand the reason for flywheels, at least in cars. A flywheel in a manual transmission car acts as an energy storage and output smoothing device. An internal combustion engine delivers non-continuous power impulses, as explosion happens in the cylinder, the force pushes the piston down, which in turn pushes on the crankshaft which results in rotational power impulse. But these impulses happen separately which results in gaps where there is no force applied to the crankshaft. Without a flywheel, your car would vibrate badly, flywheel captures these impulses and delivers continuous output.

Diesel manual transmission cars have heavier flywheels because higher compression in the cylinder results in higher torque applied to the crankshaft, therefore you need more mass to store more energy. Heavier flywheel has more rotational inertia, thus this makes revving a diesel engine seem lethargic compared to a petrol car, the heavier flywheel takes more time to reach desired RPM, thus revving a car with a heavy flywheel results in slow RPM pick up.

A fairly recent trend is putting a bit heavier flywheel to even small petrol cars, this makes them harder to stall at the expense of driver's engagement, this combined with custom fuelling maps make new petrol cars really easy to drive but not really engaging. It is common to put a lighter flywheel to cars being raced, you get faster revs and better engine response at the expense of the engine being easier to stall.

Automatic transmission cars, at least those with a torque converter, also have a flywheel, but since the design of torque converter allows for non-direct engine-transmission coupling, most of the smoothing can happen inside the torque converter itself, thus the flywheel is extremely light.