This isn't a proposal for action but rather an extended comment that flags up some things we may want to keep an eye on if this change goes ahead. I suggest distinguishing static considerations, whereby orderings of historic posts already in our repository may change, from dynamic effects this change might produce on voting on new questions. Much of the discussion, and indeed cited data, concerns the former but the latter effect is less clearly evidenced. It's well known that answers listed at the top of a page tend to receive more votes, because not everyone scrolls down to read the lower ones. This is one of the problems with treating the voting system as if it were a rating system (as @SextusEmpiricus points out).
Sometimes an excellent answer is posted weeks or months after a question was first answered, and such answers inevitably struggle to catch up with ones posted when the question had its higher initial traffic. Occasionally these answers represent changes in the state of the art, e.g. when a new paper has been published (or particularly on StackOverflow, when there are a changes to a programming language or library, as anyone who regularly refers to the
ggplot2 resources there will tell you while tearing any remaining hair out). Such answers often really deserve to be the top-listed, and a voting system will only rarely achieve this. There are ways to try to influence the votes, like high-rep users promoting the answer by posting a bounty for it and deliberately leaving it in the bounty queue to encourage more eyeballs/voters, but when a question was initially popular and the original answer garnered dozens or even hundreds of votes, it's very hard for a new answer to overcome that kind of lead even if it technically supersedes prior ones.
These dynamic considerations suggest that voting is not only dubious as a rating system, but also as a curation system. There is an advantage to having curated content. Future readers, whom this site is intended to benefit, are helped when deprecated answers no longer appear at the top of the page, regardless of vote count. Question pages "owned" by active users can be curated this way under the "green-ticked shows first" system, if superior answers come along later. This is not flawless since many of our most-viewed questions were asked by drive-by users or long-time users who have become inactive, and as @NickCox points out, a proportion of green-ticked selections seem perverse. (I think, perhaps more charitably than Nick, that some, but not all, of the unusual-looking decisions by OPs come when a question addresses their individual practical needs or clears up a point of confusion that was very specific to them, whereas a more general or comprehensive answer may be more valuable to everybody else.) I remain unconvinced that changing to a "best-voted shows first" system will solve the curation problem on old or abandoned questions due to the way voter traffic tails off so rapidly, and only gets a temporary bump when a new answer is posted. If we were concerned about curation of such questions — and I think arguably we ought to be, since many of the more important or persistent questions in statistics, the ones which readers will keep flocking to for years to come, have already been asked on this site — I think we might need more radical changes.
Perhaps SE devs could establish a "black tick" for "moderated choice" of the answer most likely to be useful for future readers, and any black-ticked answers appear first, regardless of votes or which answer was green-ticked. This would necessitate a dedicated review queue accessed by high-rep users for questions which have been flagged as having a deprecated, incorrect, or clearly technically inferior answer currently appearing in prime position. I can see obvious risks of misuse or contention in the case of subjective questions, but on a largely technical site where some answers are either objectively incorrect or outdated, and with a voting system that doesn't always correctly rate answers, some kind of solution along these lines seems inevitable eventually. This is just a piece of brainstorming rather than a solid suggestion but I can't see how our direction of travel can avoid such discussions in future, as our task becomes increasingly to curate the body of material the site has already produced. Voting in its current form doesn't seem to be a sufficiently powerful curation tool; again, it wouldn't surprise me if eventually higher voting power for high-rep users is introduced as an attempt to address the curation issue, or editing of existing, highly placed answers by inactive users becomes normalised. At some point, something will clearly have to be done to keep top-positioned answers to highly viewed question correct and updated.
If we do opt for "best-voted shows first" we need to pay attention to the dynamics of voting on new questions, and any weakening of the link between answer quality, green-ticked answer and best-voted answer. Timing affects voting not just when updated answers appear months later, but even within the first few hours or days of a question being asked, when voting activity is at its peak. Until the OP green-ticks an answer, we do operate under a "best-voted shows first" regime, and it's very clear we have a Matthew Effect ("the rich get richer") as a consequence. Answers even a few minutes earlier than another have a few more votes, get a higher position on the page, and as a result tend to attract even more votes, since not all readers/voters scroll down far enough to see the newer answers. I think most long-time users of this site will have witnessed cases where the OP influences voting by selecting what appears to them to be the best answer — at the point they make this choice, they have access to answers early voters could not see, and they are more likely than later voters to have read the full set of answers, so despite occasional perverse choices there are reasons to think their choice is better informed than the recorded vote distribution is. It often happens on questions with multiple answers that the OP green-ticks an answer which has climbed its way up to midway through the answer list on vote terms (and sometimes stalled there for a while), but after some time in prime position on the page, the green-ticked answer accumulates votes more rapidly to catch up or surpass the previously best-voted answer. If it had been left halfway down the page, I am sure it would have attracted a bit more attention and a few more votes by dint of having the green tick, but I very much doubt the effect would be so dramatic as when it gets a positional promotion too. This is why I'm not convinced data about agreement between green-ticked and best-voted answers on our existing question are a good guide to the effect of such a change in future, and some caution and a beady eye would be needed.