Wouldn't a cable that's thousands of feet long be prohibitively heavy?
A thousand foot of very strong fishing wire https://www.amazon.com/Sufix-Performance-Braid-300-Spool/dp/... is a mere 0.15 lbs, if Amazon’s description is to be believed.
So as a guide wire even with multiple strands (for redundancy) I think 3000 feet is very achievable.
But a gas turbine (a jet engine, basically) is efficient, kerosene is very energetic, and for a hoverboard I imagine the hot & compressed exhaust itself is what’s mechanically useful.
There’s no reason to have a line other than as a tether, though that comes with its own risks.