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Could you view the situation in the following way?
Consider two electrons moving side by side at the speed of light. At that
speed, each electron has an electric field but the "information" about the
changing field of one electron (due to its movement) cannot "keep up" with
the other electron. Consequently, the two electrons do not "see" each
other, even though they are (from our perspective) side-by-side. From our
reference frame, we see no force between the two electrons. We "explain"
the lack of a force by saying that a magnetic attraction exists that exactly
balances the electric repulsion. At lower speeds, the magnetic attraction
is less and the electrons do repel but less than what would be predicted by
the electric force alone.
A lot of hand-waving, I admit, but is it sound?