Sorry for the long post, but it is worth a read since i think it finds a good middle ground between Lewin and ElectroBoom

My explanations leans on this:

https://en.wikipedia.org/wiki/Electromotive_force#Formal_definitionsInside a source of emf that is open-circuited, the conservative electrostatic field created by separation of charge exactly cancels the forces producing the emf. Thus, the emf has the same value but opposite sign as the integral of the electric field aligned with an internal path between two terminals A and B of a source of emf in open-circuit condition

If you have a open circuit length of wire in a moving field (Or vice versa) you do get a different amount of charge density on each of the ends that corresponds to the EMF voltage. The longer the wire is the more electrons there are for the magnetic field to tug along so as you go along the wire they cumulatively get pushed more and more. Much like a vertical column of water getting pulled on by gravity, the bottom ends up with more pressure than the top (Yes i know water is not the same as electricity but the idea is similar). Here this effect is called charge separation.

This works even in superconductors. At first it sounds wrong because more electrons at one end would create a electric field inside of a superconductor, but the magnetic field that shoved the electrons over to the end did so using its 'virtual electric field' (Well it is a real electric field, but its not caused by a charged object). So the two fields put together are again zero. Once you connect it into a loop they are free to move so the field disappears and the current they cause opposes the outside field so then you have no electric field induced by charge separation and the field caused by the magnetic fields cancel out to zero too. The current and field sustain each other so the current flows forever and the field stays forever. Very useful for making incredibly strong magnets and is used extensively for this in things like MRI machines and particle accelerators.

If we instead close the loop by putting a resistor in series then we get a case of both. We need an electric field to push electrons trough that stubborn resistance inside the resistor so this effect of open loop charge separation on the wire puts extra electrons on one side of the resistor so they can force themselves trough using there own electric field. But because now electrons are flowing trough the resistor we have a current in the loop so the loop makes its own opposing magnetic field. So far it looks like a closed loop superconductor again, but the resistors don't allow the electrons to flow freely so they can't make it around the loop fast enough to fill the 'electron void' on the other side of the resistor. As a result some electrons are left behind on one end of the resistor and continue to experience charge separation, thus making the wire look like it has voltage and this voltage appears as a smooth gradient across the length of the superconductor. Due to the resistor limiting the amount of current the magnetic field it creates around the loop is smaller than the outside magnetic field and so the 'virtual electric field' it creates in the loop does not fully subtract out the one caused by the outside field. The field that steps in to fill the missing part is the electric field caused by charge separation and gets the sum of fields inside the superconductor back to zero as it should be.

This means that if we connect a wire between two points on the superconductor and route it in a way that generates no EMF on the wire we get current flow that is proportional to the voltage on the two points and the wire resistance.(But only if this superconductors loop is closed with a resistor in series). This gives the two points a set voltage between them that is a single value.

So lets see the definition of voltage then:

https://en.wikipedia.org/wiki/VoltageVoltage is the difference in electric potential between two points. The difference in electric potential between two points (i.e., voltage) in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points.

Wait...

Yeah this is what throws the wrench in the works.

So if you integrate the total electrical fields around the path you get zero volts inside the superconductor and all the rest of the voltage on the resistor. Since the path goes trough a different resistor depending on what way around you go you also get a different voltage. So by definition of voltage it checks out. This is why this is such a big argument, Dr. Lewin is not saying anything wrong.

So where is the problem then?

We don't have a voltmeter that drags an electron around and logs the work needed to do so. Tho if someone did make one id love to see it cause it sounds really cool. So because of this we can't measure the voltage in the exact way it is defined. What we have to do instead is tap off the voltage with extra wires and bring that voltage to the voltmeters input port. When the wires are run in such a way that they don't get affected by the magnetic field only get the charge separation effect pushing electrons trough the voltmeter so the voltmeter ends up measuring electron charge density between the points. The voltmeter becomes part of the circuit and the voltage drop on the 10MOhm resistor inside the voltmeter is this voltage we see. (Doing this adds a third possible solution to the node)

TLDR starts here:Okay our voltmeters suck... so really what is the problem?

The experiment is never explained how the voltmeter 'selects' what voltage it can see. There is no mention given to the importance of the path that the voltmeters probe wires take and why they are routed in that exact way. It just leaves you head scratching how is it possible to see two different voltages at the same point. It demolishes your intuitive notion of voltage in circuits. Many electronics engineers after university are likely still confused as to how it works.

The whole thing is explained with a schematic and using some basic circuit analysis tools. Any voltmeters in the schematics are assumed to tap off the electric field integral of the loop you want to see and ignore others. It would have been much better to explain it on the level of electric fields and electrons moving around if the goal was to show the underlying physics. Using a schematic and then simply using the ideal wire model out of circuit analysis methods and then talking about the fields inside a wire is confusing. You ether don't use the abstraction of circuit node analysts methods at all and focus on electrons in a wire (so you can interact your magnetic fields with them), or you go all the way with circuit analysis and create an equivalent circuit that shows the magnetic effects as inductors. One or the other ways of explaining it makes sense and works great! Kirchhoffs law is a circuit analysis tool and it works (for circuit analysis), its not a law that governs how the universe works.

So why does circuit analysis not agree with physical electrons moving in wires? Because that's not the point of this abstraction. The goal of circuit analysis methods is to make it as easy as possible to predict the behavior of a circuit with as little math as possible. So to not complicate something as simple as a wire it simply cuts the concept of voltage down to the effect the voltage has on components (including the effect the voltage has on a voltmeter). In this simplified world Kirchhoffs law works perfectly and because the abstraction uses voltages that we can observe in real life means that the results of these circuits analysis methods also work in real circuits with electrons running trough them.

Many simple equations you have been taught in your first few years of physics are actually set inside an abstracted world where for example the speed of light is infinite and our atmosphere is a perfect vacuum. So are they wrong? Well... in theory yes they are wrong, but they work just fine in the abstracted world. The math is much easier and faster in this abstracted world, yet when done carefully still gives results that are very close to real ones you would get in the real world.

Please use a hammer for hammering nails rather than screws.