There's nothing gentle about 38kV. You're not going to lower down a line and bleed off some charge. Once there's a viable path to ground, current will flow all over whatever is making that path, including the very air around it. The entire drone, wire, and the air it is inside of will turn into vaporized plasma as if hit by a bolt of lightning. And 38kV is low voltage in the high-tension world.

Creating current flows from a field is what induction is, so if you're imagining that you could just generate electricity from a field, well, that's what they're already doing. It doesn't take lowering a probe. You can just use an inductor.