The real power is and have always been with the council of ministers who is appointed by the member governments, and in practice only responsible to the member parliaments and not the EU parliament, who barely have authority to overrule the larger and more permanent commission, let alone propose any legislation.
Don't get me wrong I am pro EU but the current structure is pretty strange and mostly dysfunctional in trying to spread the pain evenly and prone to being dominated by large corporate interest groups.
The only difference to national governments is the lack of a right to initiate legislation. But in practice, they just stick anything they want into any vaguely related piece of regulation. They do have the power to amend.
It's correct that the commission(which it itself fairly powerless) have to ask the MEP for yes and no on directives, that is not the case on all regulations passed by the council of ministers as the council can choose to bypass the EU parliament, in cases where no disagreement exist within the council.
- no power to propose laws
- process heavily skewed in such way that it's extremely rare to throw Comission proposals entirely
- current procedure for reaching comprosive between EP and Comission is: small working group hashes out compromise text with no public oversight, is then rubber-stamped by plenary
- crucially, low citizen visibility into EP's working, no oversight
The last point is, I believe, crucial. EP is a dumping ground for politicians who failed nationally and gets virtually no press attention. What happens in national parliaments is news, covered in detail, even though the importance of EP, as the place where things can be influenced, is much bigger.
It is very much disfunctional as a democracy.
the head of the commission is chosen by the parliament, and the head of the commission then appoints commissioners
commissioners have a tendency to be disgraced former national politicians who have become unelectable at home (e.g. Jean-Claude Juncker, Peter Mandelson)
The situation is quite similar to a number of member states (e.g Germany) where neither the prime minister nor cabinet ministers have to be members of parliament and the government itself is not elected by the people.
Edit: Btw, the head of the comission is more like what the US would think of an actual president. The president, OP talks about is more like a representing nation.
And of course the national governments are elected (indirectly) by the people as well which makes everyone who they elect also indirectly elected (albeit through more indirection).
It's really no more removed than any US cabinet member, especially if you figure in the electoral college–yes, that's rather symbolic. But the difference to electing a parliament which elects the executive in the EU member state is still rather small.