what is written on the label?
Automotive electric power is sketchy, the smaller the vehicle the worse it gets, you don't want accidentally bust your test equipment.
Page1 and page2 disagree about which version is A and which is B so take care when ordering.
for those playing along at home that appears to be a CD4060 chip set-up to oscillate in the audio range, but they have pixelated the markings in the PDF.
Yeah, that should work, I'd go with battery power instead of plugging into the ATV. those things will use less power than a portable radio does, so you should get good battery life.
You can get audio jacks that have normally open switches inside them, so you could have the socket turn the power on when you plug in
If I was doing it I'd use an LM555 instead of the CD4060 the 555 has a stronger output and is probably more frequency stable, (but in the end that depends on the other parts on the board also being stable)
(555 oscillator module)
What you describe is a specialised signal generator, your specialised version is likely to be easier to use than a general purpose signal generator.
at audio frequencies where this works the only practical antenna type is a loop antenna, your buried anrtenna wires form a very large loop.
So for your wand you also need a loop. someting like several overlaid loops of fine electric wire (preferably magnet wire). maybe wound on a plastic jar lid, or on a hand reel, or some other non-metallic frame.
It's going to need a resistor in series to not overload the oscillator, about 100 ohms should be sufficient, maybe a add variable resistor too so that you can adjust the power level.