Well, no, right now average score is meaningless. But if having a higher average score means that you have a stronger voice, as OP suggested, then people will start to care about those numbers.

I assume the goal is to encourage high-quality discussions (including extended discussions/back-and-forth), in which case average comment score will work against that.

If you measure something, people will try to increase it. Just by presenting the number, people are encouraged to maximize it.

I tend to think that better discussion in a forum like this comes about with less back and forth. That's not to say that alternating comments on opposite sides of a viewpoint can't be enlightening, but I think it's stronger if those viewpoints come from multiple people on both sides.