The way I think about it is that, while individual electrons don’t flow the way molecules of water in a pipe do, the charge flows.
I refuse to watch Veritasium for a number of reasons, and I’m not surprised that his videos attract criticisms, because his explanations are generally too simple. He does tend to get things basically right, but it often tends to be a bit misleading. Thus I can’t comment directly on the videos, but I will say that in my view the most important things that flow is the information that is carried in the electric field as it changes. However, to say that electricity doesn’t flow (was that the name of a video of his? It sounds like something that I made sure to never watch from him) is like saying money doesn’t flow. True in some sense, but to hitch your wagon to the idea because it doesn’t flow like water that nothing flows at all is completely misguided, even if technically correct if you get to choose the goalposts…
I’ll contrast what he does with eigenchris , who has a complete lecture series where he develops the ideas and math of general relativity from a knowledge base of first-year calculus and high school physics. After watching his video series, I felt I actually understood what was happening with general relativity, while after I watched Derek’s video on the subject, I didn’t know anything more than what I had read as a general overview from other sources.
In the case of electricity, I’m going to guess that he’s trying to bring in ideas from quantum field theory/quantum electrodynamics at the same level as his general relativity videos, and in that case, since there are a ton more engineers who “know” how electricity works compared to general relativity, he’s going to get called out on it, even if he’s right. Those engineers don’t know the math of quantum electrodynamics because it simply doesn’t matter for them, but Derek doesn’t either. It’s pointless exercise in pedantry that doesn’t actually explain the phenomenon sufficiently, which is par for the course for Veritasium.
The reason for the two videos: many people think the mechanism for transferring energy using electricity is the same mechanism for transferring energy using hydraulics.
In hydraulics, the medium for transfering the energy is the fluid inside the hoses and pipes. Pressure and displacement of the fluid carries the energy. So a lot of people naturally think that electricity works the same way, i.e. that the medium of transferring the energy is free electrons, and voltage & current carries the energy. But that’s not an accurate description. If all of the energy is “carried” by the free electrons in the conductor, what is the form of energy? Kinetic energy? Nope… the kinetic energy of the electrons is much smaller than the amount of energy transferred. So if it’s not kinetic energy, what’s the form of energy?
The answer, of course, is fields. Electric and magnetic fields, specifically.
(If we want to be pedantic, we could say that electric fields is what’s really at play when we talk about attraction and repulsion of fluid molecules.)
Heck, you don’t even need the electrons to transfer electrical energy - they’re optional! That’s how radio waves work. But if you want to direct the field energy in a certain direction, then you’ll need metal that contains free charges.
You know what is a pointless exercise in pedantry? Writing two longish posts giving a detailed criticism of videos you say you refuse to watch. 
We had a thread on the Veritassium videos. Derek was trying to talk about fields, and not flow, but in the end tried to claim that the energy was available at the lamp in about 3 nanoseconds and didn’t go around the wires.
That was widely criticised and his second video, whilst setting the claim in a better context, is still flawed. Yes there is a small impulse of energy that is coupled across, but the full power from the source only arrives after propagation through the wires. You can make all manner of simplifying assumptions, such as the wires having no resistance, which forces the fields to the boundary of the wires, but if you model the full reality of conduction, electrons flow, and if they don’t, the energy does not propagate.
The energy that arrives later, while dependent on there being free electrons in the wires, still flows mainly through the fields outside, and along, the wires.
Statements to the effect of “electrical flow is from the fields, not the electrons” is in the category of “not even wrong” in my opinion. To a first approximation, an electron is its electric field. That is the primary way in which it interacts with the universe. If it weren’t for the field, it would probably either not exist or be some barely-observable ultralight particle like a neutrino. The behavior of an electron is essentially determined entirely by its electric field. There is nothing wrong with saying that electrical flow comes from electrons pushing on each other, because it goes without saying that the way they push is via their field.
Moderator Note
Since this is FQ, let’s focus on the actual facts here instead of our opinions about the quality of a particular post.
Thanks.
Isn’t an electron its Dirac field?
The quantum point of view is, everything is a (quantum!!) field, however you prefer to formally define them (operator-valued distributions and so on). Feynman teaches us that electrons interact via photons (i.e., electromagnetic force—more fields!)
ETA
electromagnetism includes photons (bosons), not only electrons
Let’s say I point an antenna in the direction of deep space, and transmit a signal. Would you agree the electromagnetic wave has energy, even when it’s billions of miles from Earth? How could this be possible, when there are no free electrons billions of miles from Earth?
Mobile charges, metal wires, etc. are not “needed” for electromagnetic energy to propagate. Of course, when you don’t include these things, the energy just goes wherever it wants, and does whatever it wants, which means there will be a lot of losses in the system. Mobile charges and metal wires give us a way to direct the electromagnetic field energy, sort of like a train track forces a train to move in a certain direction.
When doing RF and microwave engineering, the engineers won’t talk much about voltage, current, electrons, etc. The focus is primarily on electromagnetic waves and power. The waves propagate in waveguides & transmission lines, and the metal is just there to guide them. The same thing is true for AC power transmission, DC power transmission, and low-frequency circuits, but it’s hard to convince people of this; for this stuff, they’re convinced the energy is “in the electrons.” The truth is that the energy is in the waves for all electrical circuits, regardless of frequency.