There will be an election across all EU countries in 2024, to decide the composition of the European Parliament. The previous one was held in 2019. That you apparently don't know that indicates that simply holding an EU-wide election doesn't guarantee sufficient debate.
But you vote for the parties in your country who form a bloc - it isn't really a vote across borders in the same way as a Presidential election would be.
The Parliament is also very weak and can't propose laws, etc.
This seems pretty common in "fédérations" (or similar systems.)
I guess Californians and Texans can both vote for local politicians under the same party, but nothing force the politicians to vote together once in Congress ?
Also, I think as a bloc, the EU is not entirely ready yet for, say, French people to directly vote for non French people. But who knows.
As for the balance of powers in the EU, it's far from perfect. But coming from France, who am I to judge ?
