You make a lot of great points and your confusion is an indication that you actually DO understand the purpose of privacy to a degree.
Privacy matters most if you're creating software that other people will be able to use in their projects to achieve certain things. You want them to be able to use your code without accessing internal variables that would break things. Maybe you wrote a program that will parse text and apply correct french language accents to words (bear with me).
You want this software to be easy to use so you provide some concise, public methods that the users of your software will call to do the work. Typically these methods manipulate data in a specific predictable way. They'll probably utilize a bunch of private functions to access private variables and do the real work. In essence you're hiding the complexity behind an easy to use veil.
You DON'T want the users of your program to interfere with those hidden processes so you make them private, inaccessible from the outside.