It's exactly as you say, UK has no control on its border for european citizen, as such it's still has control of its border for no european citizen, schengen is a region for no european citizen, I misread your first text since I just woke up.
Though the difference between Euro zone and European union is fundamental, since the crisis we are undergoing is from the former. European Union holds well, the problem comes from Euro zone, and it's where you understand that a no regulation economy is kind of foolish, the idea at first was really good and supported by many economist, but ten years later, it's obvious that "weak" country like Greece Italia etc that from the beginning had no straightened economy could only fell once losing their economical sovereignty.
Now the solution comes from reforming the Euro zone, problem is to reform the euro zone which has 17 members, you need the autorisation of the 27 others EU members.
And that's why UK has slowly began to piss me off since some year, it's the sole one saying "No" at each reforms.
And don't get me wrong as I said before they have the right and the politicians must protects the strong point they have, that is the
financial sector.
But if they truly want to have only the good points without bad points, they can leave, the earlier the better, as I said yesterday, we lost years because of them.
The whole situation could be an anime, UK is the main hero, Europe is the osananajimi, and US is the new transfer students. Both are yandere so careful if you choose the other, and there's no harem ending possible.