I think you can obtain physics perspective by examining a couple of simple cases. Center of mass (COM) is a mathematical construct so an exact proof does require some algebra which is out-of-scope in this question.
We can focus on kinetic energies and collisions are irrelevant to the discussion.
First, what does it mean being in the lab frame or COM frame? We are looking at all the possible inertial frames (frames moving at constant velocity, possibly zero), and asking in which of these, kinetic energy is minimal. The COM frame is well defined, it moves at the same velocity as the COM. The lab frame, can actually be a frame moving at any other velocity.
The first simple case are two objects fixed in place. Obviously, the optimal inertial frame, minimizing energy, would be the one with no relative velocity to these objects. So you prefer the COM frame to any moving frame.
Next, consider two objects moving north at equal velocities. Again, it is obvious that your optimal frame should move north and at the same velocity (that's the only frame where energy is zero). So again, you intuitively select the COM frame.
Finally, assume two objects, one standing and another moving slowly towards the first along the x axis. Obviously, you wouldn't select your optimal frame to be moving very fast towards +X. You would not select it moving very fast towards -X either. So somewhere in between, will be the optimal velocity. The rest is algebra.