Veritasium is still wrong by his own standards because he did not give the full real explaination which is rooted in the real theory of electricity and magnetism.  Quantum Electro Dynamics.  In fact with that theory in hand the way this works out is more straight forward. He went to Cal Tech and got help from Richard Abbott at Cal tech to run an experiment on a small scale done by Dr Brian Haidet PhD of the Channel Alpha Phoenix.  This involves really running wires that are very long, and one meter apart.   Both found that some small current, mili amps, flows almost right away.  Then after the expected light speed delay over the rest of the wire the rest of the current flows.   Where Derrick Muller PhD goes wrong is as he states he does not give the full explanation in terms of the applicable field theory to make his point. 

He does this in the video below. 


Their experiment is a small scale version of this one. Where he used basically one kilometer of wire.  

After seeing Veritasium's original video but before the one by Alpha Phoenix I give my explanation.  

The completely correct theoretical explanation of why Veritasium is still wrong by his own standards. 

In short to quote "QED The Strange Theory of Light and Matter" by Richard Feynman, Chapter 3 "Electrons, page 87.  

There are three basic actions from which all the phenomena of light and electrons arise. 

Action #1: A photon goes from place to place. 

Action #2: An electron goes from place to place. 

Action #3: An electron Emits or absorbs a photon. 

End Quote

One can work out all of the math to explain how this works under quantum field theory.  Quantum field theory is tied with General Relativity for the best tested theory in all of physics.  However, as I do in my video you don't need to do that.   The result of QED and QFT can really be summed up as follows. 

The probability that an electron at one location X1 will affect another electron at location X2 if (X1-X2) squared is smaller.  This leads to the inverse square law found in classical electrodynamics with some added ingredients.  

Apply this to the battery and wire problem.  It is then less probable that an electron 1 meter away from another will be affected than an electron at inter-atomic distances.  Sum those small probabilities over many electrons and you get a tiny milliamp current he can turn on a LED bulb with.   So he's both right and wrong.  

He's wrong because he is using classical electrodynamics to explain something that may even be easier to understand looking at it in a QED sort of way.    No special software needed. 

In response to a very stimulating question/comment by the Youtube Channel EEVBlog in which it was asked.

Do you know if there has even been any experimental results confirming that energy (probabalistically) mostly flows within a conductor?
If not, do you think it's possible to actually do such an experiment? I'd imagine it would be Nobel winning stuff if experimentally proven that energy flows mostly in the conductor, as that completely trumps Poynting.
I found it hard to find any research on this at all.

This is a great question.  So I thought about it and did a bit of digging.  In short what I found was not only has this been done theoretically but also in the context of the design and operation of particle accelerators and most interestingly quantum computing.

Theoretical work:
 Quantum electrodynamics with nonrelativistic sources. IV. Poynting vector, energy densities, and other quadratic operators of the electromagnetic field E. A. Power and T. Thirunamachandran Phys. Rev. A 45, 54 – Published 1 January 1992 

There are copious other citations on this but this one is the most direct.  As the abstract says. 

In this paper, quadratic operators in these fields such as the Thompson energy density and the Poynting vector and their expectation values for specified states are found. The expectation values are, in general, time dependent. For initial conditions given at t=0 it is shown that the fields are causal, i.e., for t<r/c the source-dependent fields are zero and the quadratic operators have only their zero-point contributions. For t>r/c they have both time-independent and time-dependent terms. The time-dependent terms, though transient, are shown to obey Poynting’s theorem. The steady-state part of the Poynting vector is related to the Einstein coefficients. The corresponding electric-energy density is related to the Casimir-Polder potential for a polarizable test body in the field of the source molecules. Similarly the magnetic-energy density is derived and is used in the calculation of energy shifts. Quadratic operators referring to different field points are also discussed.

Circuit Quantum Electrodynamics. Theory and experiments:  "Multiquibit experiments in 3D circuit quantum electrodynamcis"  Jacob Blumoff  (A PHD thesis)  Circuit quantum electrodynamics in the ultrastrong-coupling regime T. Niemczyk

This is super cool because one could more or less... replace the light bulb in his experiment with a single Qbit.  If it flips then the "bulb" is "on".  To use that one  has to think about the QED of the situation.  The fundamental truth is there is no physical difference.  It's all fields.  Which I stand by being the fundamental error.  

Plus just as a youtuber ... talking about Qbits is way cooler than talking about light bulbs.  A real missed chance.