There are a variety of voting machines in use, some are like you describe. Others are ATM-like kiosks that print out your ballot based on electronic input. Verified Voting keeps a database of which precincts use which machines.

> the paper ballots retain a direct record of voter intent.

This is true, although it's expensive to recount the paper ballots and in practice people don't often do it. They routinely do a sort of checksum or sanity check by sampling small numbers of ballots. But a full-on paper ballot recount is rare.

Bush v Gore is a famous example of a recount that was halted before it finished.

A lot of the innovations and changes in US elections are the direct result of Bush v Gore, from the (ill-conceived) rush into pure-electronic voting, to modern optical scan balloting technology. You're right that Florida 2000 was fishy and super close, but I don't think it's all that much of a relevant example today. It's famous, but we've learned a lot in the last 25 years.

A better example of close elections and recount procedures is the literal tie that happened in a Virginia state legislature race in 2017. The attorneys were able to litigate the validity and intent of individual ballots in that race because they had the physical ballots, as well as the machine scans and logs. https://en.wikipedia.org/wiki/Virginia%27s_94th_House_of_Del...