We do use gyroscopic power storage, see e.g.

https://h-cpc.cat.com/cmms/v2?f=subfamily&it=group&cid=402&l...

https://www.activepower.com/

...and probably others.

(A couple of decades ago I worked for a company that was a tenant at a datacenter that used these instead of batteries; it's not new or particularly exotic technology.)