What makes you say that the ability to vote is the most important privilege? For example, I'd happily give up my right to vote before my right to legal due process. I'd rather live in undemocratic Singapore than in somewhere with free elections but widespread corruption and poor rule of law like India, Turkey, or Mexico.
Because your right to vote ultimately affects all those other rights. With enough people voting for such, protection for rights can be repealed, or even ignored outright.
Not really, since a lot of them are pretty unpopular. For example, things like "making it illegal to burn the flag" or "stopping Neo-Nazis from marching" poll very well in the US -- people would happily vote to get rid of freedom of speech if they could.
As for rule of law, there are examples of non-democracies with solidly developed rule of law, like present-day Singapore and Hong Kong, or Britain in the 19th century (and before).
You keep confusing the terms. You say "privilege to vote" in one sentence and "right to vote" in another. Constitution sees due process as a right of "persons", and ability to vote as a "right of citizens".
