Such a lottery can be had. If there is an epsilon chance that a given lottery ticket is a winner, there is also an epsilon prime chance that a winning ticket will be lost and that you will find it, for zero dollars expenditure on your part. Thus, there is an epsilon multiplied by epsilon prime chance that you will be a lottery winner without actually buying into the lottery. If you get your hopes up for this scenario, are you a free-loader?
You say you don't want to burn your opponent, but only argue against her view. Argument is a competitive pursuit. Public arguments about important subjects will likely get political. War and violence is but an extension of politics. If you have an intractable argument about a very important topic, you will have to deal with the possibility of violence.
The all-powerful tribal chieftan is a position, like an office of the President. No matter who occupies the position, it is the position that is given respect. Hunter-gatherer tribes have men of respect (tribal chiefs), and that respect is based only on personal accomplishment, persuasiveness and wisdom, because the tribes are too small for the hierarchy. So, the tribal-chief is not the all-powerful tribal chieftan. One is a man, the other is the position.
What would the chronophone transmit, if I suggested that Archimedes joined the Diogenes and the cynics?
What happens when a contemporary political and economical analysis is sent over, including the troop sizes of contemporary armies, supply levels, economic conditions of the surrounding countries, facts that could not be available to Archimedes but could be available to some of his contemporaries?
These third alternatives you talk about are too new. None of them were more probable than the immortal soul 50 years ago. Most people have not yet adjusted to derive hope from these alternatives, even if they do seem to be marginally more practical now. The other problem is that success or failure of these technologies is apparently in the hands of fallible human operators, but the soul is outside human control altogether. Perhaps, people are afraid to trust other people with their "eternal" life.