I don't think that's why it's called a "Privy Council". Just because the only meaning for privy that you know, today, is a toilet, does not mean that the word in another context means toilet, any more than the fact you know the verb "do" in English can be slang for sexual intercourse means that when Marlo says "Do it or don't" in The Wire he means for his soldiers to have sex. He's indicating that he doesn't care whether they murder Bubbles.

"Privy" more generally means private or personal, which explains its use for a group which advises specifically the monarch alone.