It does, but it would be a mistake to consider the EU fully democratic.

The European Parliament's power is quite limited in reality, and most of the power is held by the member states and the Commission in practice. It can veto some (but not all) things, but can't even introduce legislation.

And the Eurozone is definitely not democratic, it's run by finance ministers.

It does.

Some people like to call undemocratic elections that did not pan out the way they hoped.

The parliament is not in control of this; it has the power to vote directives up or down, and that's it. Crucially it does not control the ECB.

The European Parliament has the 2 powers that defines a parliamentary regime: it votes the budget and nominates the governement. Without Parliament's explicit or tacit will, nothing happens in the European Union.

I'm not an European, and I get my information from an openly biased source, but this is my understanding:

There is a parliament, but there's no real pan-European parties. So the parliament is dominated by a center-right and center-left coalition which work together to make up the government. The opposition is largely made up of "fuck the EU" protest parties who pride themselves in not participating in any sort of political affairs. And since the voting is the European tradition of "you vote for the party, not individual members," you tend to have members that toe the party lines (unlike the US political system). The end result is that European elections tend not to be able to effect any change in policy in the European Parliament.

The term I heard used for this is "democratic deficit."