why can't electrons also flow from the negative terminal in battery 2, through the right bulb, and into the positive terminal of battery 1?
Because the only way to get to the negative terminal of battery 2 is to come from the body of the battery, and the only way to get there is from the positive terminal of the battery. And the only way to get there is through the switch in the friend's house, which is open so no current can flow through it.
Batteries don't create electrons from nothing. They "pump" them from one terminal to the other via a chemical reaction, causing current to flow (through the battery) from a low potential to a high potential.
Current only flows in complete circuit means the current has to flow through the battery just as much as it has to flow through every other circuit element.
As an aside, it's also why it's silly to say that current always flows from high potential to low --- the battery is a trivial example of when it's the other way around. And it's also silly to say that all current is a flow of electrons in the opposite direction from conventional current --- the battery is a trivial and everyday example of when ionic currents should be considered.
In your question, you say 'it develops an EMF which opposes the increase of current'. That's not quite correct.
A better way of putting it would be 'when a voltage is applied to an inductor, the inductor develops an EMF which is due to the increase of current.'
We notice that if we connect a voltage to an inductor, then after some time t, a current is flowing, given by the equation \$I_t = \dfrac{Vt}{L}\$. This is the experimental observation. It is this observation that we have to work from, to derive the rules.
Inductors can seem a bit 'spooky', but the situation is exactly the same for a simple resistor. Does the voltage across it cause the current to flow? Or does the current flowing through it cause the voltage? They both happen.
We could say the change in current causes the voltage, or the voltage causes the change in current. Neither statement is 'more true' than the other. One statement may be more useful than the other, depending on what we are trying to analyse. If we are looking at a boost converter for instance, the 'voltage causes current change' is better for the inductor charge up phase, the 'change of current causes voltage' is better for the inductor run down phase.
If we start with the rules and try to work out what is happening, we will come up short, because physics only describes what happens, not why.
We know enough about what is going on with electrical phenomena to know that we don't understand it truly at a level where we can say this or that causes this or that to happen. Voltage, for instance, is a measure of the energy it takes to move charge around. To say then that voltage 'causes' something is to put the cart before the horse. All we can say is that when this happens, say current increases through an inductor, we also notice that we can measure that, which is a voltage between the terminals. Don't try to overthink electrical causes and effects.
You might like to look up Zeno's Arrow Paradox, to see what happens when you try to overthink mechanics.
Best Answer
It sounds like the solution you're looking at is wrong, because your point is correct. There is no current through AB and DC. As far as the solution you gave in the comments:
Either the solution is wrong or your schematic is wrong. The quickest way to check that is to realize that EF and HG represent essentially identical paths, but your solution has them with different currents.