He does not provide any evidence at all that the field is jumping the gap and not following the wires.
Well, it simply does. As the field builds on one wire it will quickly (at the speed of light) envelope the other wire, putting that wire into a increasing magnetic field. This will, of course, induce a current in that wire. This current will be (and depending on the separation of the wires) be much smaller than the current induced into the drift velocity of the free electrons within the wire but it will be induced very shortly after the circuit is made. Some time later the increase in the free electron drift velocity will make the trip and add to the induced current. Both mechanisms are at play.
If one used a sensitive ammeter instead of a big (useless) bulb one would see a small current induced within (depending on a reasonably small wire separation distance) a few picoseconds, then some time later (depending on the loop length and the velocity factor of the conductor) a much larger current added due to electron drift. Veratasium overstated the induction effect. He has been dragged through the fence on this.