We can infer that answerer's already disagree with one another
No, we can't. Giving a different answer doesn't mean one thinks all other answers are bad/wrong/incorrect. Possibly incomplete, but that's the only inference I would comfortably make.
It seems answerers think their own answer is better and will downvote other answers just because they want their own answer to be the best.
You'd think. But that doesn't happen. Mostly due to the 1 rep penalty for downvoting an answer.
If an answerer wants to respond to another's answer they could do it in their own answer
That's what comments are for.
As for the meat of your question:
Why are answerers allowed to comment/vote on one another's posts?
There is no good reason not to. The scenarios you have raised turn out to be non issues. They don't happen (or rarely happen).
People who answer tend to be those who are knowledgeable in the field they are answering on. Pointing out inaccuracies, bad practices and just incorrect information is beneficial to everyone. Blocking these things is not. These should be done in comments and not in "rival" answers, as this is the point of comments. Answers should not be places where one just points out issues in other answers - that just detracts from an answer.
So - there is no real upside to blocking answerers from responding to other answers and a real downside to it.