This mantra first emerged in the 1960s when proposals were introduced to modify or overrule some of the Warren Court’s decisions (on school prayer, for example.) Why then? Many constitutional amendments do not touch the Bill of Rights, of course. But even those that did or proposals that did not pass never drew this sort of reaction until around 1965.
One answer, as I discussed in my Bill of Rights book, is that the first ten amendments held a much higher status in our constitutional culture by the 1960s for various reasons (incorporation among them). But another one that I’d not thought of before was that Barnette was quoted for the proposition that the Bill of Rights should not be the subject of politics, including Article V.
Now I don’t think that’s what Justice Jackson meant. To say that fundamental rights depend on no elections refers to their restriction by ordinary statutes. If supermajorities over an extended period of time want to modify a provision in the First Amendment (say, by banning flag burning) that strikes me as valid, if mistaken. But the broader interpretation of Barnette carried the day. The Bill of Rights has not been formally amended since the case came down.