In reality, instead of 2 distrusting parties (democrats and republicans) in each polling place, there would be 100 mutually competing / distrusting services that would like nothing better than to expose the other services as frauds with indelible cryptographic proofs. That's the basis of Byzantine Fault Tolerant consensus protocols.

In the first scenario, to answer your question, if service 1 kept being faulty (saying you vote for Bob when you voted for Alice) then you'd simply increment your nonce and try another VoteChain that starts with service 52. The VoteChain determines which 3-4 services out of the 100 are consulted, and in what order. You have a few nonces, up to 10. If you claim ALL random services you've tried are faulty, then yeah, go ahead and sit out the vote, you're probably just a liar and complainer. They don't know who you are, so the chances of them being good for 100 other people and specifically not good for you, 10 times in a row, are very small. And even if it was true, that's 1 vote out of many. Now if this happens more frequently, then these services could be dropped from the 100, pending investigation -- which is easy since the services don't know who is voting, could be the police. So why would the services risk being on the hook for this?

In all your examples, you're begging the question.

In 1 and 2 in the second scenario, you assume that your own phone AND your own laptop AND all the servers all have malware and are undetectably malicious. In that case, you have much bigger problems -- they can, for example, steal money from many people, send messages to ruin relationships and reputations, and much more. In your example, large swaths of people can't trust any of your devices. In that case, society as a whole is cooked. It's not quite as paranoid as "not trusting the cryptographic algorithms and math", but it's close.

Let's assume that the Trusted Computing Base isn't compromised. Because if it is, then you may as well also distrust all the poll workers as being corrupt, and the media as reporting the wrong result, etc. After all, this system is being added ON TOP of the existing system, so it can only ADD security.

Regarding giving out private keys without revealing them to a third party, I have already said that's a strawman. They'd be giving out tokens that are used to prove that you have 1 vote, and they are put through a mixer by the people, like pulling numbers out of a hat. On the other hand, the public/private key pairs are generated by the person on their own devices (e.g. in the secure enclave). You can't steal these keys so easily, unless you steal the person's phone AND coerce them to enter biometrics when voting. But then you could just make them do a wire transfer or anything else.

Look, about this constant refrain about "coersion, defrauding, etc" this happens already. Voter intimidation can happen already, preventing you from going to a polling place, or simply disenfranchising you making it too inconvenient or far to go. It's a much BIGGER problem now, that would be REDUCED if you could vote from your phone, and on net you'd have an improvement.

Also, since in the USA you don't need to present ID while voting, a person could tie you up in your basement and go vote as you. Since in your hypopthetical world, illegal coersion and force and defrauding has no consequences apparently, then that would mean in CURRENT voting schemes, people could just vote as others.

Heck, in Australia, I could even get someone in trouble by voting AS THEM. Their name would appear twice. In Australia, they fine you if you didn't show up to vote. So without IDs, you can get in trouble either way (if you don't show up, or if you supposedly voted twice).

I'm telling you, the same people who claim IDs are totally unnecessary for voting, are the same people trying to find attacks on cryptographically secure voting. But many of these "rubber hose attacks" are already doubly possible in today's "physical" voting schemes, along with all the other downsides (the cost, the speed, the scandals, as you can see with uncertainty in elections around the world).

30% of the USA thinks that the 2020 election results were illegitimate. You can't wave that away as "well, our paper elections are great, they're just partisan hacks/deluded". I bet you with cryptographic elections, that 30% would be far less, and elections / referendums would also be cheaper and easier to do all the time. You wouldn't need to do it once every 4 years and spend billions AND it would be more reliable.

https://www.umass.edu/political-science/