There are differences with the US model, which is obviously to be expected.
A key one for me is that US president is elected by the people. The EU does not have that degree of democratic accountability. Some may think that a good thing.
But without reforming aspects such as this, or at least offering to, I think it natural that the EU will continue to face questions over its democracy legitimacy when stories perceived to be unpopular arise.
One of the problems is, to my understanding, that many people do not really understand how the EU Commission and Parliament relate to each other.
As a result, they come up with theories about how things work with "undemocratic ways" and other such misunderstandings.
In short, the Parliament is voted directly, the formation of the Commission is done with a kind of proxy (national goverments are in the loop). There is a control loop between the Parliament and the Commission; the Parliament has to OK the new Commission and if the Commission goes haywire, the Parliament can pull the plug.
As for integration, now that Great Britain is finally removing itself from the EU, we can expect more tighter structures to emerge. One of these is the European army, which has been floating around on an idea level for a long time, only to be continuously torpedoed by Great Britain. With president Trump casting serious doubts over the future of NATO and the core premise of the entire defence alliance (the idea that everyone is in it together, i.e. attack against one is an attack against all), it is only logical that Europe formulates its own pan-European defence mechanisms.
For the EU, as one of the supernational geopolitical power players, it also makes sense to tighten the co-operation on other areas, too. One of these is security. Probably one of these will be economical, and so on.
I'm not sure that understanding how the EU Commission and Parliament relate to each other will help convince people the EU is democractic. In particular, the president of the European Commission is currently chosen by some really weird ostensibly-democratic insider political trading: https://www.politico.eu/article/spitzenkandidat-jean-claude-...
Well, again, the directly voted European Parliament 1. elects the President of the Commission, 2. acks/nacks the contents of the Commission and 3. can force the Commission to resign.
I mean, it's not like a wink, wink, handshake and some random person gets installed just like that as the President of the European Commission.
The Parliament really has to approve the person, whoever it is and however the person is found.
A key one for me is that US president is elected by the people. The EU does not have that degree of democratic accountability. Some may think that a good thing.
But without reforming aspects such as this, or at least offering to, I think it natural that the EU will continue to face questions over its democracy legitimacy when stories perceived to be unpopular arise.