The body that nominates the Commission isn't elected.
In theory the Commission is mostly made up of civil servants who answer to commissioners, who are themselves nominated by each country's own government or civil service. Each commissioner has one area of responsibility only, and they answer to the head of the Commission who is their boss. So someone in the UK votes for a politician, who votes for a party leader, who appoints some ministers, and those ministers may or may not have much of a say in whoever gets nominated to be a commissioner - one of many. But there is at least a path there, even if long and indirect and the person your vote ends up influencing doesn't do anything important to your country or needs.
In practice it doesn't actually work that way. In practice, the head of the Commission has veto power over the nominations. They aren't supposed to according to the treaties but the treaties are ignored. This means that in reality it's the head of the Commission who picks the Commissioners, because they can just reject anyone who isn't sufficiently aligned with their own agenda.
So that leaves the question of how the head of the Commission is picked. Once again there is theory and practice. In theory, it's a decision of the heads of each state that they take together to select some candidates, and the Parliament then gets to vote for their preferred candidate. In practice ... nobody knows how the head is picked. Ursula von der Leyen was recently re-appointed despite being plagued by scandals and having a long career of failing upwards. Parliament was sidelined by giving them a voting list with only one candidate on it (her). Seek out an explanation of how she got this job and you won't find one because:
1. The heads of state don't talk about how they decide as a group. Is it a vote? Some sort of horse trading? Do they take it in turns? Are they even all able to take part? Nobody knows.
2. There's no record of which country voted for who, or why.
3. The process by which someone even becomes a candidate is unclear.
4. Because no head of state has any control over who gets onto the candidate list, they never talk on the campaign trail about how they will "vote" (assuming that's how it works) for who runs the EU.
In other words, the process is entirely secret. The potential for corruption is unlimited.
So when critics say the EU Commission is a bunch of unelected bureaucrats, they are right and those who argue otherwise here on HN are wrong. People who got their jobs via a process so opaque and indirect that how it functions can't be explained, not even in principle, cannot claim to be democratically selected.
She was elected with a majority - albeit not a huge one. Still: elected.
This is an example of "there is at least a path there, even if long and indirect".
How about another counterexample: In the US the members of the Federal Reserve are appointed by the President and confirmed by the Senate (so an indirect path, but fairly short), for 14 years! The Supreme Justices are appointed for life.
To take this to a hypothetical extreme, image now calling a country "democratic" where you just hold elections once per lifetime.
That doesn't really strike me as democratic, as the "demos", the people, change their minds more often than once in 14 years, or once per lifetime.
Of course, the EU I'm sure also has appointments that go beyond the standard 4-5-ish years. But my point is: the EU isn't as undemocratic as you make it to be and the US/UK isn't as democratic as you may think. Both are muddling along, and probably neither reach Swiss levels.
> The process by which someone even becomes a candidate is unclear
Your points 1.-4. apply to many appointments in the US and UK that are similarly undemocratic: To take an example from the UK: The Governor of the Bank of England is appointed by the Chancellor+PM. Again, no one knows who or why they made the decision the way they made it. Were they friends with the future Governor? Did their party engage in some horse trading with the opposition to secure other benefits in turn for nominating a particular person? No one knows.
The governor of the Bank of England is indeed not democratically elected, and people do criticize that fact. I'm one of them!
But people certainly do know how that position is selected, by whom and for what reason. The current governor of the BoE has a long history of running government financial institutions, including in the central bank itself. He is a civil servant and is thus picked by the Chancellor, who is himself picked by the Prime Minister. No mysteries there. He is eminently qualified for the role.
On 3 June 2019, it was reported in The Times that Bailey was the favourite to replace Mark Carney as the new governor of the Bank of England.[9] Sajid Javid had also intervened in support of Bailey.[10][11] According to The Economist: "He is widely seen within the bank as a safe pair of hands, an experienced technocrat who knows how to manage an organisation."[12]
Previously he served as the Chief Cashier of the Bank of England under Mervyn King from January 2004 until April 2011, Deputy Governor of the Bank of England for Prudential Regulation under Mark Carney from April 2013 to July 2016 and Chief Executive of the Financial Conduct Authority from 2016 to 2020.
You can also even just observe the following litmus test of democratic legitimacy: what percentage of people have even heard of Ursula von der Leyen (or most of her predecessors) before her appointment to the most powerful position in the EU? Contrast that with their country's president or prime minister and you will see why one is democratically legitimate and the other is not.
The current prime minister of the uk was not elected for the position, they were given it by virtue of leading the party that won the most seats, the leadership of which was not voted for by the general public.
In theory the Commission is mostly made up of civil servants who answer to commissioners, who are themselves nominated by each country's own government or civil service. Each commissioner has one area of responsibility only, and they answer to the head of the Commission who is their boss. So someone in the UK votes for a politician, who votes for a party leader, who appoints some ministers, and those ministers may or may not have much of a say in whoever gets nominated to be a commissioner - one of many. But there is at least a path there, even if long and indirect and the person your vote ends up influencing doesn't do anything important to your country or needs.
In practice it doesn't actually work that way. In practice, the head of the Commission has veto power over the nominations. They aren't supposed to according to the treaties but the treaties are ignored. This means that in reality it's the head of the Commission who picks the Commissioners, because they can just reject anyone who isn't sufficiently aligned with their own agenda.
So that leaves the question of how the head of the Commission is picked. Once again there is theory and practice. In theory, it's a decision of the heads of each state that they take together to select some candidates, and the Parliament then gets to vote for their preferred candidate. In practice ... nobody knows how the head is picked. Ursula von der Leyen was recently re-appointed despite being plagued by scandals and having a long career of failing upwards. Parliament was sidelined by giving them a voting list with only one candidate on it (her). Seek out an explanation of how she got this job and you won't find one because:
1. The heads of state don't talk about how they decide as a group. Is it a vote? Some sort of horse trading? Do they take it in turns? Are they even all able to take part? Nobody knows.
2. There's no record of which country voted for who, or why.
3. The process by which someone even becomes a candidate is unclear.
4. Because no head of state has any control over who gets onto the candidate list, they never talk on the campaign trail about how they will "vote" (assuming that's how it works) for who runs the EU.
In other words, the process is entirely secret. The potential for corruption is unlimited.
So when critics say the EU Commission is a bunch of unelected bureaucrats, they are right and those who argue otherwise here on HN are wrong. People who got their jobs via a process so opaque and indirect that how it functions can't be explained, not even in principle, cannot claim to be democratically selected.