If you still prefer an air-core transformer, you might try the following:
Instead of a “plain” transformer, use the primary coil together with a capacitor over it as a parallel resonant circuit. The pulsating DC cannot be connected directly over the primary. Instead, add a tap (wire out) at about 1/10 of the total number of turns, and connect your pulsed source between this tap and the lower end of the coil. No "freewheeling" diode needed! A first guess for the number of turns in the secondary might be about half of the number of turns in the primary. Adjust the frequency of your source so that the circuit is in resonance.
The idea behind this suggestion is the following: The coupling between the primary and the secondary is weak in a coreless transformer. This means that you need a large primary current in order to get the desired secondary voltage (and current). The current in a good resonance circuit builds up to a value that is much larger than the current it takes from the source.
I made an experimental transformer of this type. I used the components I found hanging around, without any optimisation. The main parameters are the following:
Primary winding has about 230 turns (in four layers) of 0.6mm enamelled wire on a core with a diameter of 53mm. The length of the winding is about 50mm. The inductance of the primary winding is about 2.6 mH. The tap for the pulsed source is at about 30 turns from the lower end. The secondary has 100 turns (in one layer) of 0.25 mm wire on a core with a diameter of about 45 mm. The capacitor over the primary winding is 47 nF, so that the resonance frequency is about 15 kHz.
I connected a 20 ohm resistor directly over the secondary, without any rectification. With a DC source voltage of 9V, the AC peak to peak voltage over the resistor was about 15V, so that the transmitted power is about 1.4W. It was difficult to estimate the efficiency, but it was rather low, perhaps only about 30%. It should be possible to increase the efficiency by a careful design and/or experimentation, but you have to experiment a lot. In any case, this resonant circuit-transformer worked much better than a plain transformer made of the same coils.
Finally, to your question of the use of 60Hz (or 50Hz) AC. I do not think that it is a good idea. Because the inductance of an air-core transformer is small, the idle current will be large at this low frequency.