I'll also note that nothing here is per se undemocratic. Both the Parliament and the Council are made up of elected members. The members of the Council (as members of the national governments) are indirectly elected, but elected all the same. Direct election is not a requirement for a democracy (see election of the US president or the US Senate prior to the 17th amendment or the Senate of Canada right now).
That does not mean that there isn't plenty of valid criticism of the EU's current structure, but claiming that it is not "actually democratic" falls far short of a meaningful critique.
You cannot just add 100 layers of indirection and call it as democratic as direct representatives of your smallest communal voting unit. Any mandate in more indirect position should become weaker if the only metric is indeed democracy.
Now, with what I think of as probably the ideal manageable district sizes for voters (5-7 members) that is fairly chunky proportionality, so you might still want to do MMP to reduce underrepresentation of geographically diffuse minority positions.
OTOH, there are places which have STV (usually for a whole body elected at large, but you could do the same thing in districts for a larger body) with 20+ seats in a single constituency, and if you go that big per district, MMP is less necessary.