If you solve this problem in the center of mass, each object is colliding with $\pm v$ and rebounding with $\mp v$. That is the only way to conserver energy and momentum.

Now if you (Galilean) boost it by $\pm v$, it's pretty clear one ball starts off at rest, and the other ball ends up at rest.

Now in our experience with bouncing balls: harder balls bounce higher. Note also: harder balls are harder to deform. This is not a coincidence. Once you have deformation, you are going to lose energy. You fill the ball with vibrations, they cannot all come back at once and transform their energy back to another ball without any reflection. A perfectly elastic collision is an idealization, and has no deformation.

If you solve this problem in the center of mass, each object is colliding with $\pm v$ and rebounding with $\mp v$. That is the only way to conserve energy and momentum.

Now if you (Galilean) boost it by $\pm v$, it's pretty clear one ball starts off at rest, and the other ball ends up at rest.

Now in our experience with bouncing balls: harder balls bounce higher. Note also: harder balls are harder to deform. This is not a coincidence. Once you have deformation, you are going to lose energy. You fill the ball with vibrations, they cannot all come back at once and transform their energy back to another ball without any reflection. A perfectly elastic collision is an idealization, and has no deformation.