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Notes -
IRV is simpler to tally and audit in low-tech scenarios because the votes themselves are the physical record of the count, an advantage it shares with FPTP. In FPTP, you sort and bundle votes into e.g rubber bands of 20 and boxes of 1000. You can easily verify a count by checking that a box indeed contains 1000 votes, and they've been sorted appropriately. It's easy to update your count report just by seeing you have X boxes and Y bands.
Approval voting, Score, Borda etc require you to maintain a store of the counts independent of the physical ballots, which introduces more room for human error and complicates recounts/verification. You need to increment up to N counts for a race with N candidates, and even approval voting has 2^N-1 ballot variations that complicates sorting.
If you're thinking of terms of low-tech boxes and counting processes, IRV is an intuitive extension of FPTP because you're just opening up eliminated boxes and resorting them. Practically, it's rare for this process to go particularly deep or be particularly sensitive, and the count of votes rarely exceeds twice the votes cast.
I think we're now able to have do much better than IRV and I think there are potentially clever ways you could do tear-off perforated ballots to make the counts under approval/range more reliable, but a lot of (sometimes conspiratorial) questions about the popularity of IRV miss that it was an intuitive and practical solution at the time. If Australian preschoolers can vote on schoolyard activities with IRV, I'm sure American adults can manage.
Oh, counting votes by sorting ballots never even occurred to me. American elections almost always have several races on the ballot, so that's not really a feasible way to organize the counting. The counting is almost always done by machine anyway, with hand-counting only for recounts.
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