This is a very interesting question! I guess the usefulness of anonymity depends on the situation in the country. For instance in Armenia we do not have absent

This is a very interesting question! I guess the usefulness of anonymity depends on the situation in the country. For instance in Armenia we do not have absentee ballots just for that reason. And before the 2018 revolution there was large percentage of bought/intimidated vote in all elections. There was a recording how owner of a supermarket chain held a meeting of all employees demanding from them a list of the ids and names of their relatives with promises to vote for him. With this kind of tactic their party ended up taking 68%, as many people were afraid that they could be found if they try to vote not the way they have promised. Another thing to consider is that filling absentee ballots is a much more complex endeavor, than standing at the booth and making sure people vote the right way. But in general i think in the situations when secret ballot can actually change the election result, there are much larger problems in the society, which secret ballot won't help to solve. For instance in Rome secret ballot was introduced only at 139 BC and arguably made the following period of social troubles worse. After all every vote is a dry-run for a civil war, to see who would win without going into war, and it is useful only if it convinces both sides that war is not worth it. But if someone can't even openly say how he have voted, how he can hope to protect the result of his vote? So i think in Rome it ended up causing the civil war.