I definitely agree that this is a problem. What I see that makes it specifically a problem is when one answer provides, say, solutions A, B, and C, while another answer provides A, C, and D. The answers are not duplicates of each other, but neither of them contains the 'full' answer.
To me, the ideal case is when there is one answer that contains all possibilities. That, to me, is why we close questions as being 'too broad': there are too many possible answers, so no single answer can contain all of them. The problem I see with this is that it's very easy to skip a possibilitiy, and once someone else posts it it would be mostly unethical to steal their idea and add it to your answer. Answer wikis are a possible solution, but that's more of an altruistic approach that I don't think we should expect most answerers to use.
The other option, as you mention, is to post a new answer for every possibility. To me, this seems like a great way to downvote/delete options that are wrong, or that have already been posted by someone else. It should also be easier to find the most community-appreciated solution. The problem with this, though, is that people tend to not look at anything other than the highest-voted answer, and the highest-voted answer might not actually be the one the OP finds most useful, which gets into issues with what should be marked as the 'accepted' answer.
Of the two options (list everything at once, hope you didn't miss one vs write a new answer for each), I think the latter option avoids the most issues in the short term. However, I worry that creating the assumption that for every point you wish to make, you need to make a new answer, is a dangerous move. I'd also be kind of worried about reputation farming (a two-part answer makes half as much reputation as two one-part answers). As I've said, I think a 'definitive' answer is ideal, but difficult to obtain, encouraging people to write multiple answers is only going to make definitive answers less prevalent.