In Germany, we also have paper based voting and yet, we do not have the issue that ballot boxes disappear. Everything and I mean literally everything is accounted for and if the numbers do not match up (e.g. less boxes of ballot boxes than before, or n(valid votes)+n(invalid votes)!=n(total votes) there is an immediate recount. No one leaves until shit is done, and if it gets too late everything is packed up, sealed and boxes counted.

The horror stories of the US are a failure of their system, not of paper voting.