Anonymous voting ("the australian ballot") doesn't exist to protect you from your public opinions. It exists to allow people who don't want to express their opinions in public from being coerced when they make their choice and to negate the power of bribery to influence elections. If no one can prove you voted for the right candidate, it's less enticing to hand people money to do so. American elections weren't secret in the past. This was a fix that was put in to throw a wrench in political machines ("vote for this guy or I'll break your legs") and quid pro quo fueled campaigns ("here's $100 bucks if you vote for my candidate").