I know this will come as a bit of a shock but perhaps a bit of a historical context is in order here.
When Coulomb carried out his experiment and published his findings the concept of fields did not exist, let alone electric or magnetic fields. Electricity and magnetism were accepted as completely two separate disciplines and there was no connection between the two. So, as far as the force between two points was concerned, the force acted on them and nothing else existed between the two.
The concept of electric and/or magnetic field did not happen until Faraday (a great man). Even then he could not really formulate it which happened with Maxwell (the greatest ever).
So, as far as Coulomb's law goes, it states that the force between two points-objects (in a vacuum) is proportional to the forces on each points, and inversely proportional to the square of the distance between the two points. That's it. The connection with electric or magnetic field is a later development. Now, the original form of the law is a mathematical tool for us. Replace the force in Coulomb's law with electric field strength, now you are dealing with the electric field between the two points.
So, the claim that electrons deflected by a current flowing through a wire near by is not because of a generated magnetic field but instead by simple Coulomb's law, in my opinion is wrong. Because the original form of the law also does not contain the terms that deal with the effects of "length contraction" and "time dilation" from Einstein's theory of relativity as it is also a (much) later development.
