Perhaps the down voter could explain why?

Name one current satellite that uses flywheels to store energy.

And flywheels are anything but lightweight. I've provided numbers, if you know better provide a reference.

No current satellite uses magnetic bearings, as far as I know, and while the paper linked describes a theoretical way to integrate energy storage and attitude control, few or no satellites have yet implemented this. (And, of course, if you have a functional set of flywheels, using them also for energy storage costs no extra mass. This doesn't mean it would be worthwhile to scale them up to use nothing else, or worthwhile to use them on a launch vehicle.)

@Hobbes You're right, they're not currently in use, theyre in current research. The point is, they're not a crazy idea... And I've updated the answer to reflect

Flywheels make sense as an addition to batteries, if you have large peaks in supply or demand. And they don't degrade as much with each charge/discharge cycle.

note that the question was about launchers, not satellites, i.e. a lifetime of 10 minutes to 4 hours and no recharging.

my downvote was for {{citation needed}} It still is.

For anyone interested, here is sandia's 2002 report on FESS: sandia.gov/ess-ssl/EESAT/2002_papers/00018.pdf Here is a 2017 report oin flywheel energy storage from Amiryar & Pullen @ University of London : mdpi.com/2076-3417/7/3/286/pdf

@Hobbes - I see no clear cut desire by the OP for flywheels to 'power' a launch vehicle (interesting idea). OP talks about energy storage. "Considering that the electron is launched" - IDK what that's supposed to mean...

I don't math well enough to know. "Energy stored is linear with mass, but squared with velocity - so by having a lighter wheel that can spin double the speed, double the energy is stored." @JCRM - that's the part you want a cite for, I hope?

One more description with a simpler description of flywheel operation: explainthatstuff.com/flywheels.html

@Mazura the Electron is a smallsat launcher from New Zealand. Phrases like "component of the rocket" also indicate launchers are the intended target rather than satellites. OP asks about replacing the batteries (which are used to power the turbopumps on the Electron) with a flywheel.

@Mazura en.m.wikipedia.org/wiki/Flywheel see the energy is 1/2 I W^2. I is related to mass, w is how fast it is spinning.

@JCRM still waiting for what citation you want.

no, that part is self explanatory, it's the other "facts" you put in that need backing up. To pre-empt a later objection you should also explain how an active flywheel would handle the vibrational environment of the launch vehicle.

This sentence for example needs citation: the fly wheel to be used for power grid storage is efficient because it's lightweight. Quote numbers, and compare with other storage technologies. You can even borrow the data from my answer.

@Hobbes I was just going by the maths, as said it's quite trivial. If you want more specific examples you could research the volvo S60. The question is it possible, would it be crazy .... the answer is very clearly a yes it's possible, not it's not crazy.

The S60 KERS is still at the prototype stage, and Volvo has been working on this for decades. Meanwhile, Volvo has battery-based hybrids in series production.

1/2 I W^2 can add up to a lot of energy, but you soon run into material limits: at 60k rpm, a flywheel is close to tearing itself apart. This places an upper limit on how much energy you can store per kg of mass.

@Hobbes I agree the limit will depend on the material you make the flywheel out of; which is interestingly why they're picking carbon fibre in this research ... it being light obviously means it holds less energy, but it being as strong as it is means they can spin it faster to make up for the loss in mass.