The problem is the way the institutions are set up.
Yes there's a wider argument about whether the EU is the right level of "demos" to underpin representative democracy, but even if it was you could create more democratic institutions: a directly elected President and have the legislation created by a government which was made up of Ministers from the Parliament.
Instead, legislation gets proposed by the Commission, who are appointed not elected, and then the MEPs can scrutinise it and vote on it but the is no elected government proposing legislation.
There's no process of accountability, ie you can't say well I think the European government has done a bad job so lets give another government a chance - which is what you can do in the UK for all the flaws in FPTP electoral system. The Parliament can force the Commission to resign en masse but not the electorate.