Bureaucrats? Hm. maybe, but I always thought it was more like this (but maybe I'm totally wrong):
EU parlamentarians are elected directly by "the people" in all EU countries. Problem is/has been that nobody seems to care. Just look at how many people actually vote when it comes to the EU parliament elections. A total disgrace. The result: only the very best politicians

of every country go to Bruxelles and that's why nobody knows them.
The EU commissionaries are nominated directly by the national governments of the EU countries (these governments have been elected directly by "the people"). Looking at local governments gives us an idea who the government is likely to nominate. Then let's look at how many people actually voted in the last national elections (i.e. all over the EU). Again a disgrace. The nominated commissionaries then have to proof to the EU parliament that they are qualified to perform the job before they get installed. Who were the parliamentarians again?
There are trends in the EU to not care about politics in general (resulting in vast groups of "the people" not being represented by their national governments) and this is directly reflected on the EU level. Said differently, people who are not interested in national politics will naturally be removed from EU politics., as it is designed as a supranational government.
So much for theory. And now reality:
It is true that the EU parliament has rather restricted rights for direct initiative. This has been criticised, and rightly so, I think. Parliamentarians do have some rights though (clearly misrepresented in that "feature documentary"), and even more so since 2010 , and 2014 (election of president). There even have been discussions to have EU parliamentarians elect the commissionaries directly. Britain has 73 of 751(?) EU parliamentarians. But who were those again? I think having national governments nominate candidates involves more "minds" and "people" and is thus more democratic than allowing only 73 people to decide (and who were those again?).
The influence of lobbying is worrying. But it is a problem not exclusive to the EU level.
Also, the lack of communication from the side of the EU commission has been criticised harshly. And rightly so.
And there are God knows how many public officials/officers, who earn a lot of money -- sure, nobody elected them. I'm not a huge fan of the public officer system anyway and I think this should be abolished -- both on national levels and on the EU level.
The only group of people, I think, who actually understod what the EU is, are far right-leaning and, worse even, right-wing extremists, and I feel they all want to hijack the system to establish a supranational European Reich with local hegemony

. Oh, did I just give Brexit proponents another gun? Well, since Brexit has become a topic, Switzerland is very happy, cos a lot of money from Britain is flowing their direction
