You've got it backwards: in an elastic collision, kinetic energy is conserved by definition. Removing one conservation law makes the whole exercise physically meaningless because elastic collisions require as a given that kinetic energy will not be converted to other forms of energy. This premise guarantees that the momentum is treated as a quantity separate from the total energy of the system (which, as a reply to your first post pointed out, when expressed in relativistic units, includes not just the energy-mass equivalency but energy-momentum as well).
In reality, momentum in almost every collision will be converted into other forms of energy such as when intramolecular bonds are broken and materials are deformed in a car accident or when momentum is converted into mass in particle colliders. The elastic collision equations are only useful as a rough approximation precisely because they go out of their way to simplify the energy-momentum equivalence to conservation of kinetic energy just like Newtonian mechanics is a simpler but useful rough approximation of relativity at low energies.
You can't use elastic collisions to reason about energy-momentum equivalence because their most basic assumption is that there is no energy-momentum equivalence.
In reality, momentum in almost every collision will be converted into other forms of energy such as when intramolecular bonds are broken and materials are deformed in a car accident or when momentum is converted into mass in particle colliders. The elastic collision equations are only useful as a rough approximation precisely because they go out of their way to simplify the energy-momentum equivalence to conservation of kinetic energy just like Newtonian mechanics is a simpler but useful rough approximation of relativity at low energies.
You can't use elastic collisions to reason about energy-momentum equivalence because their most basic assumption is that there is no energy-momentum equivalence.