Thank you for clarifying what you meant. Your comment is much easier to read than the article you linked to.

> if they insincerely/strategically rank begich in 1st instead of palin, then begich wins. i.e. they get their 2nd choice instead of their 3rd.

So it sounds like the real complaint here is that IRV fails the "favorite betrayal criterion"[0] which requires that "voters should have no incentive to vote someone else over their favorite".

You're right, that's a genuine flaw of IRV, and one that Approval Voting doesn't suffer from. Unfortunately, the strict definition is slightly unhelpful here, because in Approval Voting the only way to rate a candidate "over" your first preference is to not approve your first preference, which of course will harm them. So it's almost impossible for Approval to fail this criterion just by the lack of options it gives to voters.

Instead we should be considering if voters have an incentive to vote for another candidate over, or as equal to, their first preference. This criterion doesn't have a name, but it's a criterion that Approval doesn't satisfy, for the same reason as your Alaska example. Supporters of Palin have an incentive to approve Begich, i.e. give him equal support on their ballot paper to their actual first preference. You might say that this is fine, because it allows them to get their 2nd preference as the winner instead of their 3rd, but if other people had ended up voting different on the day, then giving this extra support could have denied the Palin supporter the Palin win they were hoping for.

> you can't argue about strategic voting

You're probably right that IRV has more scope for strategy than Approval, and I might even grant that Approval gets better results in practice, regardless of the amount of strategic voters, but I still think that IRV has a better UX and requires less cognitive load (as long as voters don't need to number every candidate, although that makes IRV even less fair).

My point is, as a voter going to fill out your ballot, you know what it means to like one candidate more than another, and you can ignore the candidates you don't know about, and you can probably pick an order randomly if there are two candidates who you like equally well (and randomized ballot ordering helps here, although it potentially hinders illiterate people who need to see the ballot paper long in advance to memorize how to fill it in).

By contrast, being asked which candidates you "approve" of basically can't be answered without having to think strategically, or at least it will mean different things to different people. For some people, "approve" means "I'd still be happy if this person wins", but for others it means "I'd grudgingly accept this person winning as at least they're not the worst candidate on the list". The decision for how to interpret the question will be subject not just to personal psychological biases, but candidate campaigning, and misinformation, and will lead some voters into thinking that they need to pick this "threshold for approval" based on a deep understanding of the polling data (and therefore without that, it might be better not to vote at all).

In conclusion, I don't think this is a reason not to support Approval Voting, especially where the alternative is IRV, but I do think that as electorates get a chance to use it in practice (and campaigns adjust to the new realities it brings), people might start to be annoyed at its flaws. Fortunately those flaws are smaller than those of IRV, and minuscule in comparison to FPTP, but I still think that voting reform advocates can do even better, in terms of UX and simplicity.

[0] https://electowiki.org/wiki/Favorite_betrayal_criterion