The only wool being pulled is by the EU, they are about as undemocratic in structure and in process as you can get.
We can both have our views on the amount of
democracy in the British system, but not I'm afraid compared with the EU. As your helpful article indicated the Council and Commissioners are unelected, the President is appointed and so are the Commissioners, and it is they who decide strategy and policy with no recourse to the public to change them e.g. they are not up for election every five years.
In Britain the party leaders are known before the election, the existing cabinet/shadow cabinets give an indication of who represents what brief in Government/opposition. These people, all of them, are directly accountable to their constituents and as MPs are up for election every five years.
The Lords are unelected, but like the MEP's they only scrutinize proposed policy/laws/regulations, they can recommend changes but ultimately the Commons (in Britain) the Commission (in EU) decides.
Our commons are elected the EU commission is not.. simples!