The argument that 'There are problems with our democracy, so we have no right to complain about the structurally anti-democratic nature of the EU' always seemed an odd one to me. Jorge used to trot it out rather a lot.
There may be democratic deficits in Westminster, but they are things we can vote to change should we so desire. The democratic deficits in Brussels, however, are far greater and not subject to our democratic scrutiny in any way, shape of form.
Besides, if anything, I would say that this country at the moment is suffering from a surfeit of democracy rather than a deficit.