This problem was solved at least as far back as ancient Rome. The solution was and is the secret ballot. If nobody gets to see your ballot before it goes into the box, and if it can't be tied back to you, nobody can hold you to a vote. Thus, even if someone threatens your life, bribes you, etc, the secret ballot preserves your ability to vote your conscience.
Usually bribers will want some proof, like a photo of your vote, when you go inside and there are election observers and you're not allowed to use your phone it's more difficult to provide that proof.
a) you get bribed by some external sweepstakes or ad or any which way to vote for a candidate you genuinely don't support.
b) you in your home get hit by a vandal with a specific mission to make you vote for their candidate. Remember, this felon does not know when you get your ballot nor when you voted.
Hell, which is more likely to be tracked down? The Musk trials will take months. that felon will be arrested before the week ends.
a) bribing is more probable, happened by foreign country here this Sunday
b) it's a lot more dangerous for the entire society, it's not a vandal in your home, if anonymity is not implemented by voting process, then the local gang leader will force you (and all your neighbors) to vote for him, at the date he will tell you and you won't be able to do anything as he will be in power after these elections and will make possible to fake future elections.
I find this threat model unreasonable and it reeks of conspiracy-style thinking; there are so many points of failure in it that I have trouble believing this argument is being offered in good faith. I'll continue to engage in good faith, but I want to state that skepticism.
(a) I don't think it'd be possible to extort votes in this manner in on a significant enough scale to influence an election without it being obvious that it was happening. All it would take is a few people to go "Oh yeah, someone broke into my house and held a gun to my head" to spark an investigation. Moreover, to a degree, this level of coercion is already occurring: a party platform of cutting taxes on your income bracket is effectively a bribe. A party platform that they'll cut healthcare funding if their candidate doesn't take a particular office is a gun to your head.
See also: the publicly known instances of 'vote bribing', e.g. Ben and Jerry's offering free ice cream, or Musk or Cards Against Humanity offering money/tickets/etc to people who have voted/make a plan to vote.
(b) for a vote-extorter, how would you verify that I voted as demanded? what if I don't mail that ballot? what if I deliberately screw up my signature? what if I submit my ballot before you extorted me? what if I vote in person later, invalidating my mail-in ballot? It's possible someone with access to the inner workings of the voting judgement process could detect some of these, but at that point, again, it's already compromised.
(c) If you have the resources to possibly (1) identify vote extortion targets (2) successfully threaten them in such a way that it would influence their vote (3) validate that they did indeed vote as you've demanded, you surely have the resources to attempt a more directed attack on, say, the mail system or the vote system itself. Trying to extort individual citizens en masse would radically increase the odds of detection of your operation.
(d) how do you know your election judges aren't under gunpoint? how do you know they haven't been bribed? Or your county clerk?
Yes, in most cases coercion isn't as obvious as someone literally pointing a gun to your head (though in some cases it could be, there's nothing to prevent it). Typically it'll be something more subtle, like the "vote bribing" scenarios you outlined, but with the added pressure of the person providing the incentive literally being able to stand over your shoulder and watch you fill out the ballot, then take the envelope from you and deposit it themselves into a drop box.
As I've said before, coercion doesn't need to be overt to be effective. Just a small amount of social pressure applied over a large number of people is enough to make a significant difference. That's why typically there are laws banning campaigning right outside polling places. Now what if the "polling place" is the entire country, over a period of multiple weeks? How are you going to enforce that?
Consider also that the electorate being able to trust that elections are free and fair is nearly as important as them actually being so. Its not enough to just say "that's probably not happening at sufficient scale to make a difference"; you need to be able to convincingly demonstrate to voters it actually isn't. Having a system that's robust to these types of cheating schemes (as in-person, secret paper ballot elections have been for centuries past) is the best way I know to do that.
