They were a thing in Formula1 car racing, very briefly.
There also were buses that were powered by that, those were called Gyrobus.
Flywheels are limited in capacity to hold energy by material limitations. There are still people researching to improve fly-wheels although those are more like magnetically levitating rings now. The ultimate fly wheel is a stream of magnetic particles in a giant linear accelerator that loops back onto if self.
Today super capacitors have enough power to service many applications that used to be flywheel territory. If you could have a bicycle with a fly wheel power-storage (or capacitor bank) that would allow you to go up and down hills while you only have to push the peddles hard enough for the average power consumption, that would be awesome. (I'm assuming that's what your picture shows)