A typical situation is that there's a contentious issue, and some anecdotes reach your attention that support one of the competing hypotheses.

It is interesting that you think of this as typical, or at least typical enough to be exclusionary of non-contentious issues. I avoid discussions about politics and possibly other contentious issues, and when I think of people providing anecdotes I usually think of them in support of neutral issues, like the efficacy of understudied nutritional supplements. If someone tells you, "I ate dinner at Joe's Crab Shack and I had intense gastrointestinal distress," I wouldn't think it's necessarily justified to ignore it on the basis that it's anecdotal. If you have 3 more friends who all report the same thing to you, you should rightly become very suspicious of the sanitation at Joe's Crab Shack. I think the fact that you are talking about contentious issues specifically is an important and interesting point of clarification.