I've noticed the opposite, using a different driver with adjustable frequency and duty cycle I found that for a fixed duty cycle going lower in frequency allowed the secondary voltage to be much higher. With just 12v and a frequency in the hearing range I could get a continues arc that jumped a 7cm+ gap whilst at 24v it was well over 10cm but just made the LOPT flash over. The MOSFET was a 650v .03mohm type that could take some ridiculously high current, I made sure to keep the peak drain voltage below 500v as they were about £12 each!
OK, so what you're seeing is most likely normal flyback behavior.
More on time --> more average current draw. At constant D (and assuming saturation never occurs in the transformer), power output is inversely proportional to frequency.
The waveform likewise shows increasing peak voltage following turn-off (when current peaks), but since it's less frequent, it's not as powerful as it would be simply increasing D instead.
In class E, the peak voltage typically isn't too much greater than the supply voltage. Ideally, you'd have fixed off-time and adjustable on-time, which has the effect of increasing D and decreasing F at the same time -- giving better control over the output.
Primarily, though, you'd want to control a class E driver by varying its supply voltage.
Maybe you don't need/want an output that hot -- that's fine. Then you need some way to adjust off-time much longer. Which, you really need a different circuit for, it seems.
I've also driven them in half-bridge too where resonance was used as a form of FM slope detection in combination with an audio signal injected into the RC timing section of the drive IC, it actually sounded pretty good and loud as a plasma speaker!
This was using LOPTs from CRT TV's, the PC monitor ones didn't work nearly as good and the massive internal capacitors would "snap" over if the voltage got too high. Internally what makes the monitor LOPTS different from the TV versions? Are they driven differently inside a monitor?
Possibly a combination of lower voltage ratings, with high frequency construction. Instead of one big winding (as the old CRT FBTs had), it's wound one layer at a time, with a HV diode per layer. This is done so that each layer can have the same AC voltage along its length -- so that inter-layer capacitance doesn't pile up, severely limiting the self resonant frequency.
HV tracks tube size, usually, so a big (30"+) TV might use 30kV, while a 17" monitor might use only 20kV.
Tim