Just like, that’s such a bad system , why did people put up with it?
Because hindsight is 20/20 and people had preconceptions back then that filled in the gaps, as they do right now.
The gaps are and were actually full of nonsense like “he’s my buddy I’ll give him money” but people expect the process to be a lot more reliable and solid, because they think they’d be more careful with that kind of money, not realising that to some millions are pocket change (and nobody is careful with pocket change) and that others gamble with other people’s money and thus are a lot more cavalier.
The reason there is no such thing in elections, is to prevent vote buying/extortion.
In Italy it’s such an extreme problem that any ballot where the party is not marked with a cross on the party logo and (if present) a block capital name next to it on the provided line, is automatically discounted, because stuff like writing a name a specific way or using crosses, checks, dots, or other symbols was used to track vote buying/voter intimidation in mafia controlled territories.
Some vote counters and polling station overseers would be on the take and keep track of if the votes they expected to see showed up when counting ballots and report back.
If you were able in any way to prove something beyond the equivalent of an “I voted” sticker it would immediately be used to ensure people voted a certain way or to exact some sort of backlash on those who didn’t.