Just to better understand the question: What do you mean by "the European Commission should be democratically elected"? Unless I'm mistaken, it already is. The President of the Commission is for instance chosen by the European Council (where the democratically elected Heads of Government represent the member states) and then confirmed by the European Parliament (where the democratically elected MEPs represent the European voters).
I don't see how this is structurally different from (or less democratic than) other parliamentary systems (e.g. Germany, UK...) in which the Head of Government is elected by parliament. If anything, it's even more democratic as the candidate is proposed by the member states and not by a party.
I meant the Commission would then be elected directly by the people much as MEPs are; no degrees of separation. It would be kind of similar to the 17th Amendment in that respect. Appointed officeholders should hold legislative power, even if they are confirmed by the people's representatives. It's the same reason the Lords can't draft legislation themselves.