Germany hand counts their ballots, results in quickly

[Igel]

45. Gerrymandering doesn't matter for US presidential and Senate elections, either. n/t

As for the # of polling machines, that's something that's widely discussed here in Houston.

There's a limited number of voting machines and polling locations. They compromise in how to distribute them in a given election between voter turnout in the last election at that location + how public opinion polls state the likely turnout for each community. If it's an election of great importance to Ethnic Group A but not Groups B-E or the communities Near the Bayou that Flooded Last Year but not other communities, for instance, they may put their thumb on the scale and send a few extra machines to Group A or "Near the Bayou" dense areas.

Have paper ballots, the number of polling places isn't limited by the amount of equipment. (But then counting's a nightmare.)