Why is the EU undemocratic?

Because it was set up to try and avoid future wars by pursuing integration, not to represent the will of the people.

AFAIK the EU Parliament cannot even propose legislation. The unelected Commission holds that power.

You have to remember there are layers of governance here. The EU is a conglomerate formed of democratic member states. The Commissioners are appointed by those individual governments (which have to be democratic, it's a requirement of EU membership, for some value of "democratic" anyway). They also have to be approved by the EU parliament.

So we have Commissioners who are appointed by elected governments (in much the same way that the UK government appoints ministers, that's one level of indirection away from the voters) and are approved by directly-elected MEPs (assuming that the UK's shambolic turnout rates for EU elections mean the MEPs are actually directly-elected, since most people don't seem very interested). So the Commission is, what, half a level of indirection away from the voters? And they're directed by the governments they represent to adopt certain positions and advocate for what is in their government's interest.

So, yeah. We don't directly elect the Commissioners, but we don't directly elect the Chancellor of the Exchequer either.