Euhm. Sort-of. The commission is appointed by country governments (specifically by the minister of foreign affairs). That minister is appointed by a government, which tends to be a coalition government, chosen by the head of the executive, that then has to be ratified by a majority of the national parliament of the country. An appointment to the commission remains valid until their term runs out, no matter what happens to the government that placed them there.

So it's EU commissar (appointed) -> Foreign Minister of member country (appointed) -> Coalition government (appointed) -> Head of the executive of member country (ie. the king, in most cases) -> Parliament of member country (elected)

Certain member countries don't have very stable coalitions, which means there's always a few EU commissars that don't have the support of their host governments. This is especially true now that Christian parties are in the process of being voted into irrelevancy in a lot of countries (unfortunately to be replaced by rightist and in some cases extreme-right parties)

So, what you said. With one very important difference: there isn't one level of indirection, there's at least 4 or 5 levels of indirection. It isn't very democratic, especially since EU law means these people have law-giving power, power to override the parliaments of member countries (that would be why Christine Lagarde proposed using that power to freeze wages)