Batteries are needed when there is no sunlight. Flywheels would be more vulnerable to temperature changes during day/night cycles. I am not saying it wouldn't work though. It may indeed be a good solution.

My own research shows that flywheels are excellent backup power sources for factories & offices that would suffer financial loss from brownouts, or temporary blackouts. For example if the power goes out for 5 minutes the machines would shut down. If they are used to mix chemicals for example, and the process is temperature or time-dependent, then a blackout would require that the systems be cleaned out, re-initialized, and re-started. So a 5-minute blackout could cost 3 hours to get the machines running again. If it happens late in the day, that means sending all the production staff home.

For spacecraft I don't know if a flywheel system would be worth it. But I think flywheels are used for guidance and propellant-free pointing of sensors, telescopes, etc.