In relating momentum to force, you have force = (change in momentum) / (change in time) which is equal to mass x acceleration (the definition of force is mass x acceleration). Acceleration is equal to 1/2 times the velocity squared. That's why the use of 1/2*m*v^2 is applicable to this debate, and m*v is not.
Choice of conservation of energy or conservation of momentum are both seemingly valid. These type of particle physics problems are traditionally done with conservation of momentum. Open any physics book and see for yourself.
A more interesting example is connecting charged capacitors in a certain way (I won't explain). You get two vastly different answers depending on if you use conservation of charge or conservation of energy. Turns out conservation of charge is the correct way, the excess energy is lost via radio waves. That's why you need to back up conceptual physics with experiments (as I did).
Anyway your logic for why you must do things how is completely skewed. Force is actually the time derivative of momentum. It is only defined as F=m*a when the mass is constant. If you want to talk about something like a rocket, you need to talk about linear momentum and define F = d/dt (m*v).
But of course you probably know that. I only teach this stuff for a living and last I checked, each student pays $112/hr for my services. But what do I know...