This is, of course, not an easy thing to do. A lot of the judgments to be made are difficult. And we can always offer guidance before enforcing via deletion. However, I see no way around it, so I think it's a tough path we are forced to follow if we wish the site to flourish.
So how would this look? I've tried to think of some deliberately vague examples. As with everything in life, things may vary depending on details.
- Probably good: an answer based on direct personal experience, with an explanation that makes it clear this is the case. ("I've dealt with A before, and I've found that doing X helps achieve your goal.")
- Probably good: an answer based on indirect personal experience, which is clearly applicable/extensible to the case at hand. ("I haven't seen precisely A, but I have seen B, and I've found that doing X helps achieve your goal.")
- Probably bad: an answer that provides arguments but not experience, especially if it reaches a conclusion that differs strongly from those based on experience. ("Obviously if someone wants A, they also want B, and so you should do X.")
- Probably bad: an answer based on opinion/belief. ("It's just common sense that you should do X.")
We should also keep in mind the context and potential for harm when flagging. If people's feelings are getting hurt, that's a sign we should be increasingly cautious about unsupported answers. On the other hand, if it's a low-stakes situation that no one has direct experience with and people are tossing out potentially useful ideas, we can probably be a bit more flexible.
And this is about mitigating an issue, so getting even catching even 50% of the bigger problems is already a good start, and could even be effective about shifting the general site attitude and expectations about answers. We don't have to try to hunt down every last instance (especially on older questions), and we don't have to try to apply this strictly in cases that are extremely unclear.
To take a step back, I think that it's really valuable for us to provide guidance and support for the moderators here. They're the ones who are ultimately going to have to handle our flags, and in some cases delete upvoted answers. In order to do that, they need to have meta posts/policies to stand on, and they need to be able to look forward to support when they're inevitably challenged.
If we agree about this path, it may be that we eventually need to discuss some specific examples to help get this started, and we will likely need to discuss some after the fact when challenged. If and when we do so, let's approach it carefully. We do not want to put people on trial for their answers.