I had just been writing "Perhaps Brian himself will chime in with his reason(s)" when his answer appeared.
I think the system works well the way it is. One this site and the others that I frequent, multiple answers are rare (I recall noticing only three in the many thousands of posts I have read). The reasons for posting multiple answers could be
Is this answer different to the other answer?
It could be that the author has completely changed their mind (not the case here I think). In the current case the second answer is different enough.
Does the other answer have history that could be invalidated by updating it?
This could have been used by Brian (but he does not report that) because the previous answer was accepted. In another case I have seen extensive pro and con commentary that would be invalidated by an update.
Could it gamed? How would that work? One might imagine that multiple answers might give the author more votes, and it works in the cases I have seen because the answers were different. And that's OK.
But if a person posts an answer that is very similar to a previous answer, users often vote it down, regardless of the author. To me, this is the key. The community rewards the behavior it likes, punishes behavior it doesn't.
So my conclusion is that there is no need for change.