> But even when the USA sees the light on proportional representation, there will continue to be winner takes all elections. Executive positions (president, governor, mayor), senators, judges, ballot issues, etc.
Maybe, maybe not. Most of those don't fundamentally have to be single-winner elected things, e.g., executive positions don't have to be single-winner systems; multiseat executives do exist, after all -- even many US states have multiseat executives, but do those by single-winner election with seats allocated by portfolio (see, e.g., California's "Constitutional officers".)
> In those cases, approval voting is the correct answer.
Approval voting is, IMO, generally a poor choice for most elections (its good for elections in groups where disapproval or approval is tied to some substantive commitment -- e.g., if you are voting on where to go to for lunch, and "approve" on an option means "I will go if this option is chosen".) It doesn't make a lot of sense for most other purposes (because "approval" doesn't mean the same thing across different ballots -- this is actually true, though to a far lesser degree, of systems that require a forced ranking ballot, which is why I prefer ranked voting systems that can use unforced rankings) -- for ballot props that potentially might conflict, a system that merges features of majority and approval makes some sense (that is, it takes a majority to pass, but in the case of conflict the larger majority trumps the smaller one) and is, in fact, already used in some places (e.g., its what California already uses for ballot props.)
> Easy to explain, easy to tabulate, most mathematically fair, hardest to game.
IRV isn't much worse on any of those, and is if anything more familiar (and if you are using STV for multiseat elections, IRV is identical, so even though its a little harder to explain on its own, if you are using STV elsewhere, IRV for single-seat elections is pretty much free in terms of complexity, either of explanation or administration.)