This argument that everyone's vote or at least the default method of voting MUST be absolutely inviolately private and secure while our most personal conversations and details of our lives be matters of government scrutiny seems, to put it in the most charitable light, "inconsistent".

A much more plausible explanation would seem to me to be that making voting more accessible regardless of security implications will change the voting demographics in ways that haven't been vetted and agreed upon by both parties in our two party system.