To take it one step further: how does the election authority convince everyone that their vote was counted?

Us "computer folks" convincing ourselves is the easy part. Maybe after enough studying of the system, I'd be absolutely sure that my vote was counted, but what would the other 90 % of the population think? This not a "people are stupid" argument, a brain surgeon or nuclear physicist or philosopher isn't going to put in the effort to understand fully homomorphic cryptography, hardware-based security, etc. Why should they?

The alternative works perfectly fine and it's so easy to explain. It's obvious to anyone, no matter their level or field of education, how their vote makes it from their hand into the tally, what prevents it from being misused and what they can do to further reduce that already tiny probability.

You can see, feel and experience the technical measures keeping your vote safe. The people keeping their eyes on the ballot box alp day may literally be your neighbours, as are the people counting. You can reason about their motivations, you can theorise in your head about what lengths you'd have to go to in order to change even a few votes and how unlikely you would be to succeed.

The biggest danger to elections, ultimately, isn't the risk of cheating. It's the risk of losing the people's trust.

(I'm in the local election commission (that's layer 2 out of 4 in our particular election management hierarcy), we just had a big referendum, everyone keeps asking me about online elections so this rant was fresh on my mind)