Easier at a scale of "national grid for n hours" and beyond than batteries. Much harder than piling up unused coal for a decade or two, noticeably harder than the equivalent stash of hydrocarbon gases or liquids.

For these energy quantities and consistencies, wouldn't a flywheel or localized hydro storage solve for n hours? (Next to a local reactor obv)

Flywheels are quite limited in their storage capabilities. Because there is a limit to their RPM before they tear themselves apart from centrifugal forces. They are really interesting for energy storage on the scale of seconds, to smooth out spikes over that range. Much less interesting for storage on the scale of hours to days. Where batteries and hydrogen become more interesting. Pumped hydro is also quite nice on those scales, but cannot be implemented everywhere.

Hydrogen is much less efficient that batteries, but much more scalable. That makes it attractive for taking in large amounts of over-produced electricity, and for use-cases where you need hydrogen anyway rather than meaning to turn the hydrogen back into electricity.

There is one variety of flywheel that evades this problem. The magnetically confined kinetic energy storage ring levitates and confines a rotating ring via magnetic forces. The mass of the magnets and tunnel required scales in proportion to the radius, while the energy stored scales as the square of the radius. This does not violate the virial theorem, since the design is taking advantage of the underlying bedrock to carry the confining load.

It's a rather whacky idea, and as far as I know it hasn't been looked at for decades, but it is out there.

https://digital.library.unt.edu/ark:/67531/metadc173309/

Yes, in underground caverns, like natural gas is stored. This is a demonstrated technology.