I applaud @cfrank 's effort on the chart, even if I might disagree with the conclusions. As of right now, I don't entirely understand the chart, but that's okay. I don't entirely understand Arrow's, Gibbard's or Satterthwaite's theorem's either, and I've appreciated their effort for years.
I generally agree with @rob with respect to IIAC. I agree with him that it's absurd how cardinal-voting advocates try to declare victory and say that cardinal methods pass IIAC, and appreciate the example he provided in a different comment (@rob's comment
"For instance, if there are 3 candidates, and I approve Alice and Bob, but not Chris, for it to be independent of irrelevant candidate Chris, they have to assume that I would still approve both Alice and Bob if they were the only candidates."). In general, single-winner elections are asking voters to do one thing: pick a single winner. There is no way to avoid the comparative/competitive aspect of single-winner elections. Trying to escape the clutches of electoral impossibility theorems by saying that ratings of candidates are independent of the other candidates being considered is rank silliness (pun intended).
Generally, it seems wise not to think too hard about a specific impossibility theorem, and just assume that any good table with binary assessments of criteria are going to have "FALSE" or "NOT PASSED" or whatever for every method for at least one criterion. Election methods are graded on a curve, and each table is a different professor. Arrow's table only had four or five columns for the different criteria. Other tables have more. The tables on English Wikipedia are arbitrary and capricious (as many professors are), and seem to be rather moody with respect to pass/fail criteria. Note that when professors grade on a curve, it means that the student is in competition with other students in the class. It seems that rankings are pervasive.
Even if we agree that some tables (or some "professors") aren't very helpful, I hope we can agree that perfection is unattainable in election methods. We shouldn't spend too much time arguing about specific tables as we each try to gain personal insight on different methods. @cfrank - I suspect that your three meta-critieria ("Stable", "Simple", and "Consenual") are a bit too vague to be useful to newcomers to electoral reform. I'm not sure that there are three meta-criteria that work, but please, keep trying to find them! As we all know, it's almost impossible not to get lost in the weeds when trying to understand election methods, and a good visualization could be helpful.