The Commission election involves several steps.
First, the President is nominated. While this is technically a power of the European Council (not to be confused with the Council of the European Union), in 2004 the Parliament won the right to determine the president and the nomination by the European Council is only a formality. (Similar to the appointment of the British PM by the British monarch).
In practice, the choice of president is now the result of the elections to the European Parliament through the so-called "spitzenkandidaten" process [1], where each group in the EP nominates a candidate and the candidate of the largest party that can command a majority in the EP is chosen, who is then formally appointed.
Second, the Commission President in conjunction with the member states nominates the remaining commissioners. The EP conducts hearings, after which the Commission is voted on. The EP can reject the Commission. In 2014, this resulted in the rejection of the Slovenian commissioner [2] and the Hungarian commissioner being stripped of the citizenship portfolio [3] as the result of questions regarding Hungary's human rights record.
The EP can also remove the Commission through a motion of censure, Article 234 TFEU. This is what happened to the Santer commission (except that they resigned before the EP could vote them out of office).
Note that under the British system, the Prime Minister and his or her cabinet does not even have to face hearings. The members of the cabinet are simply chosen by the PM, the PM is appointed by the monarch.
That only the Commission can initiate legislation is not much of an issue and has more to do with how complex EU legislation is, which has to conform to the legislation of all the member states. The EP can propose legislation through Article 225 TFEU, can attach legislation to other legislation (such as the budget) through amendments if necessary (similar to how the US Senate gets around the Origination Clause) and simply force the Commission to initiate legislation or block other legislation (or, in the worst case, vote the Commission out of office).
[1] https://en.wikipedia.org/wiki/President_of_the_European_Comm...
[2] http://www.irishtimes.com/news/world/europe/slovenia-s-nomin...
[3] http://www.novinite.com/articles/164253/Hungary's+Navracsics...
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
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.
This byzantine system, so thoroughly removed from the populace, is one of the reasons I'm not sad we're leaving.
It is not. Even if you want to dispute the fact that the Commission President is de facto chosen by the voters and elected by the EP, the European Council has zero role in the selection of the remaining commissioners, who are nominated by the member states in consultation with the Commission President.
I'm not sure how you can live with the British system, though, if this is such a problem for you, as it is even worse in that regard.
Where did I say I can live with the British system?
It reeks worse than the EU. I'll vote out of Westminster when I get the chance (which will never come for Hampshire, I'm not a fantasist)
The Commission then originates all legislation, which Parliament can approve or not.
It's not a system I would buy into voluntarily.