To be frank, it is a quite eccentric way to appoint an executive leader: imagine POTUS being chosen by state governors.

> She is no less democratically elected than the prime minister of a parliamentary republic

Well I would say this is arguable. Sure, her power is ultimately resulting from a popular mandate. But, if your country elects the head of state directly (say, France) then there are two "hops" between a popular mandate and the head of the EC. Whereas, if you are in a parliamentary system, it's three "hops" (MPs -> head of State -> EC president).