I did indeed misread "president" as "minister", but I don't agree with you that "indirection" (i.e. appointment of officials by elected officials) is "not a lot of legitimacy". It may not be "a lot of directness", but I'm not concerned about this any more than I'm concerned that, e.g. police officers or judges are appointed, rather than elected (certainly in the places I've lived).

Edit: I don't agree that there is as much indirection as you say, either. You brought up two examples of interim governments where ministers were not elected. You haven't given any examples where this happens in er, ordinary? governments and I don't believe there are any in the EU. But I'm happy to be corrected if I'm wrong.