HF/HV - rectification or modulation?
I've come across the discussions where some engineers believe that the air core resonant transformer can have super high efficiency of transformation/energy saving potential

Adding resonance to an air-core transformer makes it more efficient than it would have been without.
However normal transformers are close to 100% efficient already, so developing a chain of converters in the search for even higher efficiency is doomed to failure.
If you are seeking better than 100% efficiency, then dream on. We don't do over-unity on this site.
Tesla coils are strictly for
a) where you want lots of pretty sparks (sole purpose of 99.9% of all Tesla coil builds)
b) where you need high voltage and some isolation between input and output, and can tolerate the losses and inconvenience of this sort of transformer.
If you want the rectified output to be 12-24 V eventually, then it would make sense for your Tesla secondary to be a few turns of wire loaded with an appropriate capacitor for resonance, producing the ballpark of your target voltage, rather than many turns producing a high enough voltage for the pretty sparks effect.