No, private keys are never handed out. Also there is no database of public keys, just the blockchain storing the transactions allowing a vote per public key.
Most people don’t trust crypto systems. They ignore them.
Most people use crypto systems. They can't ignore them. Or do you think there is anyone who asks their bank for an unprotected connection?
Well now that you’ve said correct we are in complete agreement
Yes we are. Part of my job is to make crypto systems work whether or not people "ignore" them. But if people don't trust my work, then they won't hire me, so it something I have to worry about.
There MUST be a database of valid voters and there MUST be somebody saying “you there, yes you are in this database and therefore may use your key to vote”. Any system that doesn’t do that is just letting any random jackass with a computer vote.
HTTPS != using a crypto system.
This isn’t a crypto system that people could ignore. AND it won’t solve ANY of the problem. You’ve already admitted that.