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The basic problem was that the mathematics differed between the case of a conductor moving relative to a magnet vs. a magnet moving relative to a conductor. Einstein used the Lorentz transform to transform the experiment into a frame where the electrons are not moving, then transforms back to get the result. In this way he can dispense with the concept of magnetism. So far so good.

He also mentions this. You have one clock mounted on the earth and another on a moving railway car. According to the Lorentz transform the clock on the moving car moves more slowly as the car moves away. If the car turns around and returns at the same velocity, then the clock on the moving car also moves more slowly, so when the two clocks meet again at the same place then the railway clock will be shown to be slower.

All right, but the situation seems symmetrical to me. If we are seated in the railway car then the same argument applies, and the stationary clock should be the slower one. So each clock is slower than the other, which can't be. I've also been exposed to that argument about the parallel mirrors used as a clock, but that has the same symmetry so the same difficulty is there. If I am on the railway car looking at mirrors mounted to solid ground then as far as I'm concerned the light travels farther between those mirrors and the earthbound clock will be slower than my clock on the car.

By example it is implied that the clock in the frame that is always inertial will be the faster, but I don't see why this should be short of appealing to experiments.