No system is immune to bad faith actors undermining trust by just making shit up.

The idea that if we had paper ballots with walk-in only, ID-verified votes, that Trump and Fox News et al wouldn't have launched a coup attempt, or wouldn't have gotten nearly as far, is pretty laughable. In fact we have Trump on the record saying as much: "It doesn't matter whether you win or lose, you have to fight like hell."

Let's take your scenario. In fact there is a way to verify which vote was correct (and it's used already) which is to match the signatures between the voter registration and the ballot.

This is sufficient except in cases where the registration itself was fraudulent, which would have required 1) having someone's ID at time of registration and 2) intercepting the mail sent to their ID- or utility-bill-confirmed mailing address.

An attacker who could achieve that could also have achieved just walking into the polling place to achieve the same false vote.

Any attack is extremely expensive to do at any meaningful scale, and all scaled attacks are detectable by simple statistical sampling.

If meaningful amounts of tampering were detected by statistical sampling then you can obviously re-run elections with tighter controls, more burden, lower turnout, and a generally-less-representative result, which is why we start with a more inclusive system. This can and does actually happen. Whether it's necessary is literally 100% statistically knowable through basic sampling.