I'm not from the U.S. but as my country's elections work the same way, I feel compelled to weigh in on this. Here in the UK, you go to your local polling station, you give your name, they check it against the list, then cross you out and hand you a ballot. (This was tweaked in the last few years to require government ID, but the process remains the same. More on that later).
While it's true you could in theory say you were anyone on the list, you'd have to first know you were picking a name that wasn't going to be used, or hadn't already. This is already something of a reach. If someone uses a name that had already been used, or someone turns up later to vote and finds their name crossed out, it's going to set off alarm bells.
On top of the logistical challenges, this is a high-effort endeavour. A single person going to multiple polling stations repeatedly doesn't scale super well. Obviously you can try and do this en masse but the more people are involved the harder it would be to keep secret. If you're trying to rig a local council election with low turnout, it might make a meaningful difference. Does it work if you're trying to swing a congressional race or higher? I see the mentions of carousel voting, and am aware of the likes of Tammany Hall, but these are more of an open secret. What the likes of the GOP are alleging is that there's an invisible epidemic of voter fraud to engineer distrust of the system generally.
Sadly in the UK our long-established voting system was tampered with by the government of the time, who took a leaf from GOP voter intimidation and suppression tactics and mandated government-issued ID at the polls to solve a an almost non-existent issue, leading to tens of thousands of eligible voters being turned away at the polls. Thankfully this moronic and clear abuse of process is likely to be reverse before our next major election, however.