FairVote’s biggest issue with Approval Voting appears to be what they see as a vulnerability to bullet-voting, brought on by LNH failures. They use examples of bullet-voting in their articles (and I’d love to see your refutations!) to show how, with extremists running, a few naive moderate voters get the system repealed.
I think the key to understanding why this isn’t an issue in most high-stakes elections is to realize that cardinal systems don’t require voters to approve multiple candidates to work well. Yes, you heard me right. I think the biggest benefit in cardinal systems is that they give the voters total leverage over the candidates; no longer can one hog the nomination of an entire party and chain all that party’s voters to picking them; instead, just the simple threat of being able to switch to another candidate while also backing the old frontrunner does the trick. In fact, I suspect a great deal of bullet-voting will happen, for the simple reason that a voter doesn’t care enough to express their support for nonviable candidates. By changing the frontrunners and atmosphere, you already are eliminating the need to vote for more than one or two candidates as well. That’s the greatest thing cardinal systems have to offer.