I finally watched this video through and it really had me questioning my understanding of electricity. It actually took some time for me to wrap my head around the theory of his version on "how electricity really works".
Waking up today, I have a different idea. If his theory is true, then the size of the wire doesn't matter because it's not the part that carries the energy. If that is the case, why do fuses blow with too much current and other issues such as "ohms lay" stated above by apogeum.
I don't buy into everything the video is saying. You still need a large enough wire/conductor for a certain amount of current/energy. He is completely right when he states "the electrons generated at the power plant are NOT the same as the electrons you are using at your house power outlet". Current is transferred using electro-magnetic force. This is the same electro+magnetic force that is transferred using magnetic energy in transformers (where there is absolutely no direct wire connection between input and output).
However, part of what he is saying could have an effect. There "could" be "magnetic fields" outside of the wire and these magnetic fields could be what "pushes" the electrons forward that flow across the molecules of the metal wire itself (essentially electrons inside the wire). ==BUT== the current is still being transferred inside of the wire, which means the type and size of wire is still a critical element of how the current is transferred.
I'll bring up a previous conversation I had over in the diyaudio.com forums. There was a question where guy was asking if it was worth it to do a balanced interconnect. I remember reading something like this elsewhere, so I stated balanced can be better because the cable has 2 conductors where the exact same signal is being passed, but in inverted polarity/directions. I also stated that this created a "hysteresis" between the two conductors. I basically got called an idiot and was laughed out of the thread. Ugh, okay. Understandable. Pretty much 99% of the crowd there is "if you get measure it, it doesn't exist".