>the President of the Commission is still nominated by the European Council

Yes, technically correct. What happened last time however was that the EP said "We want Juncker and we'll veto anyone else". There was a long staring contest and the EP eventually won. A veto isn't a technicality that you can approximate away - it's just as important as being able to propose laws.

The national leaders generally didn't like this, because they prefer the EU being more like a (huge, weird) deal negotiation table than something with direct representation. The national leaders are ultimately in charge, so this whole episode is mostly an unforeseen consequence of past treaties that they're willing to tolerate.

UK with its house of Lords, no constitution, monarch as head of state, and members of parliament who somehow seem to come from same schools/universities is not exactly the best example of Democracy

But hey look at EU over there

The U.K. Has a constitution, just not a single written document. It's the oldest extant constitutional government: even at the time of the Magna Carta 800 years ago the English were following constitutional principles.