I know some people (maybe you aren't saying this) claim that Gyrators don't store energy - but as far as I can see this isn't the whole story. Gyrators do store energy. They store it in the capacitor. They store less energy than the equivalent inductor, but they can still store a lot of energy.
As I understand, a capacitor stores the energy in the electric field between it's plates, the inductor in the magnetic field. The key difference is the way in which the energy is stored and then released.
An inductor stores energy when current flow generates a magnetic field. It then releases the energy when the magnetic field collapses back down and in doing so it induces a voltage in the coil. The voltage is mostly limited by the load impedance, so it can rise to a higher voltage than the supply rails. This fact is exploited in tube amps. So, energy goes in via a current flow, comes back out as a voltage that drives a current through an impedance.
A capacitor stores energy when a voltage is held across it's plates, which drives a current to accumulate charge on the plates. It then releases energy when the charge flows out as a current through an external load. The key difference here is that the discharge current is voltage driven so the voltage can not ever exceed the supply rail that was used to charge the capacitor in the first place. So the gyrator can never properly mimic an inductor even if it can store a good deal of energy.
Or maybe I've got this wrong