It is easy to get confused about about this, and it is best to think it through in reverse order to your questions.
Although any "time variation in the current" certainly produces radiation, there is more to Askaryan emission. Askaryan radiation is not just "similar" to Cherenkov radiation, it is coherent Cherenkov radiation produced collectively by the moving pulse of relativistic electrons in the shower.
This radiation is only coherent for wavelengths longer than the effective size of the moving excess charge pulse.
For such long wavelengths, individual electrons in the pulse are not resolved and the "charge distribution radiates like a point charge" of size $Ne$, where $N$ is the number of excess electrons and $e$ is the electron charge.
Shorter wavelengths "suffer from destructive interference and coherence is lost". Since Cherenkov radiation is proportional to the square of the charge, this coherent radiation is proportional to $N^2$ and dominates over short wavelength incoherent radiation that only scales as $N$.
The effective size of the shower depends on the relevant particle interaction lengths in the material, and "to good approximation the (electromagnetic) pulse is the fourier transform of the spatial distribution of the excess charge".
In ice, the shortest relevant size is the Molière length ($\sim0.1$ cm) and the longest is the hadronic interaction length ($\sim 1$ m), so without getting into complicated calculations and simulations we expect coherent radiation will only happen for wavelengths $\gtrsim0.1-1$ m, i.e. radio frequencies $\lesssim 0.3-3$ GHz.
Since Cherenkov radiation power increases with increasing frequency, we expect the radiation spectrum to peak at these radio frequencies.
This hand-waving is only order-of-magnitude, but is consistent with actual Askaryan ice measurements and calculations.
