And it has a major problem: There is no European public. Cultural differences ad language barrier make it hard to follow debates and issues. It is a lot simpler to follow my elected governments behavior.
Also the parliament would lose its style of working. Currently there is cooperation accross parties and a less strict "government vs opposition" than in most other parliaments, which means that MEPs actually got a vote (in the areas where the parliament matters) instead dof being whipped by party leaders.
And then: Most decision power is with the council, which is made of democratically elected governments (if we ignore the Hungary problem ...)
With it's 24 languages the EU debates have interesting interpretation challenges, as they don't have interpreters for going from any language to any language, but often the translate first into one language (say from Latvian into German) and then some other language (German to Portuguese), which loses a lot of nuance and color from the language.
Also media can cover it better, with few languages and politicians can provide their press statements in those few languages.
And then culture is a lot more similar, which helps to identify the "relevant" topics and way to talk about it.