This very close, pretty much about equal to the election process of the US senate before the 17th amendment in 1912.[2] The difference being no popular vote for the president. As is the case for the "head of state" for pretty much all European countries. Exceptions are only France?
The major difference is that only the commission has the right to legislative initiative. To balance this the EU parliament can dissolve the commission based on a vote of no confidence.
So all in all, it's a truly democratic system with all the checks and balances. The major difference is the lack of cross European parties, thus your choice becomes collected together in the same basket as what your chosen party thinks is the right path. Completely through representative democracy, either from the head of government or European parliament.
[1]: https://en.wikipedia.org/wiki/Council_of_the_European_Union
[2]: https://en.wikipedia.org/wiki/Seventeenth_Amendment_to_the_U...
[3]: https://en.wikipedia.org/wiki/European_Commission#Dismissal
[4]: Kurzgesagt - Is the EU Democratic? Does Your Vote Matter? https://www.youtube.com/watch?v=h4Uu5eyN6VU
Then, suddenly, in early 20th century, the federal US government made a huge power grab, and now the states are mostly just subjects of it, with much less sovereignty remaining. That’s what some EU skeptics are concerned about as well: sudden sweep and becoming subjects of Brussels.