Then the British PM is also unelected, as is the British cabinet. Heck, it's even worse, as they're directly appointed without the House of Commons even getting to vote.
> 2) it doesn't strike you that this convoluted process with several confusing and in some cases similarly named governing organizations is itself the problem?
I am not particularly enamored with the names myself, but the process is not particularly convoluted in practice. I was spelling out the details in an effort to avoid technical quibbles.
The moment the EPP won the EP elections in 2014, it was pretty clear that Jean-Claude Juncker would be Commission President, even though it technically involved a couple more steps, which mirror the steps that other countries have, too (for example, in Germany, the Chancellor is also technically nominated before the election by the President and thereafter formally appointed, but as in the EU, these are purely formal steps in practice).
And let's not get started with the election process for the US president.
The election of the Commission is not more complicated than the election of the American Cabinet. One could do away with it, of course, as Britain does, but that would reduce democratic legitimacy, so I don't see the point.
2. British cabinet ministers can also come from the House of Lords and not just from the House of Commons. This is rare in modern times, though the Leader of the House of Lords, a cabinet position, always is a member of the Lords.
2) it doesn't strike you that this convoluted process with several confusing and in some cases similarly named governing organizations is itself the problem? It gives enough of a shade of the franchise to call itself "democratic" while insulating itself from the consequences through technicality and obfuscation of responsibility