Some DCs do, but the need to keep them spinning 24/7/365 is a big maintenance issue, so given the advances in battery tech, I'd assume batteries would supplant flywheels for cost and safety reasons.
For the most part, but every few years the bearings will need replacement. This means shutting the entire thing off and dismantling it for several hours. For redundancy, you therefore need multiple flywheels to cover for the maintenance periods.
Batteries can and indeed do go bad without warning, but current sophisticated UPS monitoring systems are able to detect failures, track battery ages and raise alerts (whether anyone actually acts on those alerts or not is another matter...). I've also found that lead-acid batteries last longest as long as there is some amount of power continuously trickling through them - I have an APC UPS at home with its original batteries from 2010, and they still provide ~1h of runtime at 20% load. And with most DC-grade UPSen, batteries can be replaced in situ without powering the device or any connected equipment off.
So is keeping batteries up to date. Many a failure has been attributed to stuff like:
- batteries gone bad without anyone noticing over the years (lack of acid, crystallization, loose contacts, dust in cooling components, ...)
- switch-over between grid and battery inverters fails somewhere
- some part of the inverters fail when they're idling for years and then have to go into full load suddenly
A huge flywheel only needs bearing lubrication, that's it.