That is not the case.
The 1957 Treaty Establishing the European Community contained the objective of “ever closer union” in the following words in the Preamble. In English this is: “Determined to lay the foundations of an ever closer union among the peoples of Europe …..”.
> The root question: how did an organization that ushered in things like the Euro become a body that decides whether Europeans are allowed to have personal privacy?
Sensationalist framing aside, how does any government become a body that decides anything?
Powerful people get together and decide that they know what's best for people. Then they claim that there is "consent" because people are given the right to vote and that there is a "social contract" that no one actually has signed, which everyone should still abide by.
But the takeaway from this shouldn't be: "screw the EU", it should be: make the EU more democratic, and give more power to the parliament and less to the backroom machinations of member states. That's exactly what the pro-EU reformists want to do. Or you could pass an EU Constitution that enshrines basic rights including privacy, which the pro-Europe activists tried in 2005 (it explicitly mentioned communications privacy) but failed due to anti-EU pushback and fears over "sovereignty".
That, and (somehow) enforce the basic principles of subsidiarity and proportionality, which they are supposed to do already. That would go a long way towards not misusing that centralized power.
I will have to read up on that 2005 event, sounds weird to me that countries would complain about there being constitutional rights at the EU level. Not sure how those rights would conflict with local ones. Unless there were positive rights, like "the right to internet" or the like, which would be ridiculous and not what I'm proposing (just basic negative rights).