What you lose in mass could be made up in velocity and radial distance. Depends on the breaking strength of the fibres. I haven't done any calculations myself to see if they are more promising than a steel one. A carbon fibre flywheel exploding might be somewhat less dangerous than steel if the bits flying away can crumple dissipatively.
Regarding where can they be used... they are just batteries with a different form factor. You can put them on the grid to have some inertia and be a place to dump or extract energy spikes. They may be commercially viable if rare earth based batteries become very expensive. At a smaller scale one could use them as a mechanical UPS for a building/datacenter. Maybe even to power golf carts, not sure how well they will steer because of the angular momentum.
That was somewhat relevant for data centers years ago when inverters were much more expensive, and even then only for the 20 seconds it took to start the diesel by connecting a simple soft starter to the diesel's induction generator (bringing it up to 50/60 rps (1500/1800 rpm) in about 5 seconds, then waiting for the turbo to spool up before it can deliver full power).
Even on the grid, batteries for sub-hour duration storage are cheap, as long as you place them at an already existing AC/DC converter site like a solar plant (or a modern internally-DC datacenter's centralized grid rectifier (AC/DC converter)).
Or even a HVDC transmission line.
Or a sufficiently modern aluminum/zinc smelter.
Pretty much anything large enough to bother that has at least a boost-PFC on the input. Because with those you could just put them there, beef up the capacitor a bit or better yet, use a native 3-phase PFC that doesn't have strong 100/120 Hz ripple on that capacitor, and then literally just control the already there input transistors to do your grid jobs. If it was the very cheap low efficiency rectifier approach, it also needs the rectifier upgraded to be controlled, so just use it on the higher efficiency ones before upgrading the others. (It's like 1% efficiency difference on 240V, and 2% on 120V power supplies.)
Flywheels can have much higher power density than any kind of battery.
There is no competition between batteries (low power density, high energy density, low storage cycle efficiency) and flywheels (high power density, low energy density, high storage cycle efficiency).
Flywheels (preferably levitated in vacuum) compete only with supercapacitors and superconducting rings (SMES = Superconducting Magnetic Energy Storage).
Supercapacitors/flywheels/SMES have their high-power applications, for which batteries are not appropriate.
Using them where batteries are the right solution is of course not a good choice.
Regarding where can they be used... they are just batteries with a different form factor. You can put them on the grid to have some inertia and be a place to dump or extract energy spikes. They may be commercially viable if rare earth based batteries become very expensive. At a smaller scale one could use them as a mechanical UPS for a building/datacenter. Maybe even to power golf carts, not sure how well they will steer because of the angular momentum.