What is the purpose of a flywheel on a magnetic generator?

Flywheel on a magnetic generator's main purpose: maintaining the energy of a magnetic generator system
In fact, with today's electronic technology, there is no need to "maintain" the mechanical type. It is maintained by electric oscillation, the energy supplied is radiant energy. See details here: Self-sustaining Generator: Magnetic, Zero Point Energy, Nikola Tesla

Flywheel energy storage (FES) works by accelerating a rotor (flywheel) to a very high speed and maintaining the energy in the system as rotational energy. When energy is extracted from the system, the flywheel's rotational speed is reduced as a consequence of the principle of conservation of energy; adding energy to the system correspondingly results in an increase in the speed of the flywheel.
Most FES systems use electricity to accelerate and decelerate the flywheel, but devices that directly use mechanical energy are being developed.
Advanced FES systems have rotors made of high strength carbon-fiber composites, suspended by magnetic bearings, and spinning at speeds from 20,000 to over 50,000 rpm in a vacuum enclosure.
Such flywheels can come up to speed in a matter of minutes – reaching their energy capacity much more quickly than some other forms of storage.

A Flywheel Energy Storage System with Active Magnetic Bearings

A flywheel energy storage system (FESS) uses a high speed spinning mass (rotor) to store kinetic energy. The energy is input or output by a dual-direction motor/generator. To maintain it in a high efficiency, the flywheel works within a vacuum chamber. Active magnetic bearings (AMB) utilize magnetic force to support rotor's rotating shaft without mechanical friction. It also makes the rotor more dynamically controllable. A prototype of FESS with AMBs was developed. Dynamical model is obtained and analyzed for the rotor-bearing system. Control method is determined in accord with the dynamical characteristics of the flywheel. AMB's parameters are obtained by parameter identification. Influences of the magnetic force on the nutation and procession of the flywheel rotor, and of the controller to the stability of the dynamical system were analyzed. Experiment has been undertaken. The flywheel has steadily past through its flexible critical speed and reached to the rotating speed of 28500RPM. Maximum tip speed is 450m/s. Maximum electrical discharge power reaches 40W. Discharge duration is 100 minutes.
If you're learning how to install magnetic generator, here's a solution to help you: How to Build a Magnetic Generator