Parent could be talking about the scanning and tabulation. In theory, there shouldn’t be anything problematic with “you vote on paper, and then slide it through a scanner, from whence it goes into a sealed bin”. Then you can still hand-recount.
There are many races to vote in at the national, state, and local level. Besides President, every ballot would have included house rep and 2/3 of states had a Senate race. Then add state governor, house rep, and senate which often align on the same cycle. Most states also elect attorney general, treasurer, education and several other statewide executive offices. Some states elect state supreme court judges. Then there can be mayor, city council, school board, port commissioner, and other local offices, although many states align these to non presidential election years. Then there are states like California that always have several initiatives.
In practice, you are looking at dozens of races on this year's general election ballot in many states. Counting all those on paper would be time consuming and possibly less accurate than using an optical scan counter.
A typical U.S. ballot may have 20 or 30 issues on it.
Even if one assumed that all other factors were equal (they aren't), even the raw counting might be expected to consume 20 or 30 times as much time.
We vote on a LOT of things here, and it becomes quite complicated.
A U.S. voter may well be voting on issues related to the sewer district, and the school district, and the library district, and the city council, and (this is important) those various districts might be overlapping or even disjoint. Just because both Bob and Alice are in Sewer District 5 doesn't mean they're both in Road District A. Far from it.
It takes time just to ensure that everyone has been given the proper ballot, much less count and combine the figures from various polling stations (Sewer District 5 may cover multiple polling stations, as may Road District A, but they may not be, and probably aren't the same stations).
It's a mess, granted, but we prefer voting on stuff rather than electing a council or MP who takes care of everything for us.
The moment you introduce any kind of electronic system into the voting or counting process nobody can understand it all anymore, there's a million lines of code and billions of transistors, nobody can validate the entire thing and even if they could, you couldn't, you'd have to trust the people can.