I don't see how one is more democratic than the other.
I'm not saying that national governments should give up sovereignty. I'm saying that what powers the EU already has should be more directly democratic.
If your country's Council member doesn't represent your interests, that's not the EU's fault, that's your government's fault.
Literally every person who gets to do anything in the EU Parliament, Commission or Council is ultimately in that position because of a democratic election in that member nation -- be it via the direct vote for the Parliament, the national election for the head of state or the combination of those two for the Commission.
The EU isn't full of corporate cronies because the EU sucks. The EU is full of corporate cronies because the member nations' governments suck. And that includes the UK (which in the EU has been at the forefront of various things people like to complain about -- including in the UK itself).
Even assuming the citizens of member states were fully conscious that this was happening, who do you think would be appointed to those positions? Who would you say sucks in this scenario, the member state governments or the EU?
But we're not talking about slush funds for doing nothing, we're talking about who makes policy. And funnily enough in Westminster we also have a Cabinet and PM setting the policy agenda that are chosen not by some universal franchise but nominated by the representatives of different localities. The transfer of executive power from the MP for Witney to the MP for Maidenhead happened without the involvement of a single person outside the political class, and I didn't vote for any of the Bill Select Committees proposing new legislation or any of the individuals heading up any of the departments responsible for implementing it. And so yeah, I'm holding Conservative MPs rather than the public or the drafters of Britain's constitution responsible for how Brexit gets implemented
Weirdly, few of the people insisting that the UK must leave because of the terrible system of government the EU has are remotely bothered about similar democratic deficits in Westminster.
The UK was a major driving force behind many of the international trade agreements that are being criticised for infringing on the rights of EU citizens, for example.
Do I need to point out why that argument is flawed?