I'd agree with this entirely, honestly. (Including the sleight-of-hand I'm engaging in to make things seem exactly the right amount of weird; see footnote 14.)
Infinity is so weird that it's really hard to agree on what infinity "should" look like. And the axiom of choice only matters in situations where the weirdness of infinity really kicks in.
So yeah, some people look at the whole list of AoC equivalents and think they all seem pretty reasonable. And other people look at the list and think none of them seem that plausible. And a lot of us are split, and find some of the claims obviously true and others obviously false.
One of my goals in this piece was to try to let everyone see both sides of this: why you might find the axiom compelling and why you might find it troubling. And then to offer a pragmatic resolution at the end, which is basically another take on what you just said.
I think the article artfully uses a framing of the problem that hides the inherent, inexorable alienness of infinite sets under a familiar-seeming surface just long enough for it to come bursting out in the solution. It's a trick of timing. If your brain lingers on the problem long enough, you realize that the problem itself violates our ordinary intuitions, independent of the solution. The idea of an infinite number of people agreeing on and memorizing representative members of R^N from each equivalence class and then determining the relevant equivalence class by reading an infinite number of real numbers from an infinite number of hats is completely fantastical. How could any solution to this problem not be nuts?
More to the point, if the problem is presented without a bonkers framing that already violates our intuition, is the solution from the Axiom of Choice still counterintuitive? I didn't find it to be so when I first encountered it in class. Maybe the intuitiveness depends entirely on the metaphor used to bring intuition to bear.