Earlier quoted context omitted.
The EU is governed by the European Commission, which is not elected. Say what you will about reactionary British conservatives, the fact remains that the EU is not a particularly democratic organisation.
How is that true, if the body that nominates the European Commission _is_ elected?? By the same argument you could say UK or US or any other solidly democratic is not democratic, because some commission or organisation is not directly, by the people, elected. (If you go for the direct election argument, the UK fares pretty badly BTW.)
It's a matter of degree rather than a binary. Representative democracy is a little less democratic than direct democracy. Elections every 20 years are a bit less democratic than elections every 5 years. Having the elected representatives appoint a head of state is a bit less democratic than electing one directly. The more layers of indirection you add, the more it becomes a bureaucratic oligarchy.