Did you see my response to the parent, where I give a sketch of how conservation of momentum follows directly from conservation of energy in all frames of reference? What is your objection to that argument?
Your sketch takes as a premise, something that is actually a consequence of conservation laws: the way quantities change under frame-of-reference changes.
You can get both by looking at the least action principle under frame of reference changes and using Noether's theorem, but that's not what your sketch is about.
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.
Fundementality is a different thing, but it's more natural to assume conservation laws and take frame-of-reference equations are a consequence. You can do it the other way (I'm pretty sure, but not 100% sure because my physics math is rusty) but you're still making assumptions, and you still don't have one conservation law following from the other.