Flywheels only make sense on a massive scale. We should put one in space.
The benefits are mixed. Amongst other problems, the Earth's own rotation actually puts a lot of stress on the rotational axis. free-floating flywheels aren't generally highly practical, so compensations (I believe magnetic bearings are common) are used.
The apparently obvious storage mechanism ... turns out to be rather difficult to operationalise.
It'd be fun in a scifi movie to see large disc shaped buildings sticking half way out at an angle all over.
Most grid-support gyros are in fact cylindrical rather than disk-shaped, as this optimises for mass at the outer radius of rotation. To gimbal-mount a large cylinder you'd have to have a proportionately large spherical enclosure.
I'm only somewhat generally and vaguely aware of the issues here and I don't know what R&D or theory are specifically ongoing here.
As noted, polar and equatorial locations would require the least adjustment, as polar flywheels could be aligned perpendicular to the ground, equatorial ones parallel (along the north-south axis), which would minimise any movements, as others have noted.
(I don't think that Earth's axial inclination would matter, and am presuming that precession of Earth's own axis is Too Small to Matter.)