A counter-argument to my claim that momentum is kinetic energy is that momentum is vectorial, while kinetic energy is scalar; and therefore, momentum cannot be kinetic energy.

The fallacy of that argument is that kinetic energy has no requirement to be scalar. In evaluating collisions, kinetic energy follows vectorial distribution as much as momentum does.

If squaring velocity makes ½mv² scalar, and there is no way to do a vectorial analysis with velocity squared, then this is a third proof that ½mv² is not kinetic energy, because collisions require vectorial distribution of kinetic energy.

As I show in many places on this web site, the requirements of kinetic energy are the same as the requirement for momentum. What kinetic energy does is the same thing as momentum does. The characteristics of momentum follow the characteristics of kinetic energy far better than do the characteristics of ½mv².