How do electrons choose their path?

To speak of quantum state, the electrons should be coherent, have a non-negliglible de Broglie wavelength. On the other hand, a non-zero resistance imply a nonnegligible dissipation and is sure to break any coherence. Your pure quantum state is then quickly turned into a classical probabilistic mixture, and your electron behaves exactly like a water drop in a river which separates itself into to branch in front of an island: its trajectory depends on its position inside the cable, relative to impurity.

Edited :

To be more quantitative, the quantum effect can roughly be seen on a length of the scale Λ, where Λ is the thermal de Broglie wavelength, given by : $$ \Lambda = \frac{h}{\sqrt{2\pi mkT}} \simeq 10^{-11}\mathrm{ m} $$ where the numerical application is for an electron at room temperature. The quantum effects are therefore already negligible at the nanometric scale.

Think of it as a queue of people trying to enter a building with two entrances, both the same size and length. As people push each other, and as the people on the front move, queue proceeds, and statistically on equal amount in both entrances provided they are identical.

Same principle applies for different R's, R and 2R, where one entrance is smaller than the other one.

As for your argument. It is correct mathematically, in my opinion.