Current really isn't flowing through the air. The relevant equation for resistive materials is J = sigma E; J is current density, E is the electric field, and sigma is resistivity. Air's resistivity is huge - if you pump any significant current through air you will cause arcing.

What really happens when you transmit energy through air is charge accumulation. Think of a parallel-plate capacitor - electrons accumulate on one side of the plate and holes on the other side. If you draw a black box around the system, it looks like current is flowing through it. But no significant current is actually going through the dielectric, or you will ruin the capacitor.

Electrical engineers model the phenomenon that Veritasium pointed out as capacitive coupling. In a circuit diagram, we would literally just draw an additional capacitor in between the relevant circuit elements.

In DC, this doesn't really matter after a certain settling time because the capacitor has settled to a certain charge. But in AC (or DC right after you flip the switch) it is non-negligible.

Edited to add - to be clear, there IS an electric field in the air - but the current density is negligible unless you've caused dielectric breakdown.

Those videos are misleading. Electrons flow inside the wires. They just flow much slower than what you would expect. When you close a circuit, the information that it was closed goes very fast but the electrons flow slowly.

(It happens also with water, if you have the shower with only very hot water and you open the cold one, the output of the shower changes almost instantly but you need like a second to get the mixed water with that is warm.)

What flows outside the wires is the energy. It's very unintuitive but it's true. Feynman has a nice lecture about it. But note that most of the energy flows very close arround the wires, a very small part wanders far away.

There is an exception when you have a radio transmissor with an antena and a reciver. Then the energy flows just through the air (or vacuum). Also when you have a light lamp or a laser.

Actualy every electric circuit emit some radiation as a bad radio tranmisor. But most of the times you can ignore it.

The more straightforward analogy with water is a pipe that is completely filled. Even if the water moves very slowly, the fact that it has started moving is immediately detectable as some of it will start spilling from the other end of the pipe right away. If you have a turbine installed at the other end, it will also start spinning right away. Moreover, if you make the pipe wide enough, even very slowly moving water will move a lot of volume in a short period of time, and thus transmit a significant amount of energy to the turbine.

"are really fields per Maxwell equations" ignores quantum mechanics. Einstein got his Nobel prize for showing the fields interact as quanta such as photons and electrons. You can see the electrons doing their thing if you put them through a cathode ray tube as found in old analogue oscilloscopes

That Veritasium video has been completely debunked by people dedicated to studying and especially working with electricity. Derek is a hack and you should look for real science channels, not popsci slop.

What's a good source that explains the truth then?

I don't want to go screen videos right now, but his is vague and misleading about what it means for a bulb to "turn on". Putting current into a wire will also put a temporary tiny current spike into any adjacent wires. But that's it until the main current gets all the way around the loop.