The key observation is that, since counting votes is an inherently distributed problem (with a comparatively simple centralized step at the end), you can always deal with it by adjusting the number of polling stations.

I can speak for what happens in Portugal. We use the d'Hondt system with paper ballots, and it is not uncommon to have around 15 candidates on a ballot in certain elections, though we have no write-ins - only one checkbox per candidate.

In the last elections there were about 4,000 polling stations. Since about 6,000,000 people are allowed to vote, this is around 1,500 people per polling station on average (obviously, the distribution is not uniform). Turnout seldom exceeds 50%, so in practice the number of votes is much smaller.

Votes are counted by hand - no automation at all - at each polling station. Usually, within about 5-6 hours 99% of the votes have been tallied, with the remainder done with by the morning after.

I would say it is demonstrably workable to count votes by hand, even with a large number of candidates. I concede that write-ins may present a difficulty, but honestly: since (afaik) in the USA you can only vote on designated candidates, how difficult can it be to have all of their names appear on the ballot?