I'm not talking about the 'regulation' of the output voltage which is function of loading, not input voltage.
Yep, air-core transformers are not very good for this application. That was one of the things that I found while building my high school science fair exhibit (I built a step-down transformer) I could drop the voltage output to nearly zero as I gradually withdrew the metal core. Since I ende up having to wind a total of 5 coils during design and testing...I got pretty good at counting the turns.
