If ther parliament has no initiative right for legislation, there is no directly democratically legitimized legislative body. And the EU comission is way to indirect to be considered properly legitimized.

The parliament can order the comission to draft a law regarding a certain area, and then amend it to their heart's content. It's indirect, yes, but as said I think keeping the external drafters is actually a good idea. EU law is complex.

That the US is lackluster in its criminal system should not be an acceptable example for us.

The issue with the US is not structural, but that the enlightenment never arrived among the people there.

While your citizenship extends to allowing you freedom in other EU countries there is still plenty differences outside the EU. But you wouldnt find it that inside a nation state people from one federal state can travel under different rules than people from another federal state. They are all subject to the same rules and that needs to be the case for the EU too, to achieve statehood.

Well the EU is trying to make things like visa rules of other countries uniform for all EU citizens. Being a single state would certainly help with that but I don't think it's a prerequisite.

With the unified administrative system it is not about control from one place. It is about uniform standards and uniform interactions. For instance getting married requires different papers and prerequisites depending on country. Things that are a complicated act with multiple personal appointments in Germany can be done in 5 minutes digitally in other countries. This is unacceptable for a nation state.

It is the norm in Germany: Want to leave a public church? In one state you do that at the Standesamt, in another in the courthouse. Administration is state prerogative. Interoperability is a good thing but telling member states how to do their administration is a no-go, it's a breach of subsidiarity.

So we are still a long way from having the EU works as one nation state.

Agreed, however I'd still say that the lack of a European voting public is the main issue. The rest is details which will work themselves out or actually irrelevant. And the voting public thing is necessarily a generational project. If the whole electorate was Gen X and younger things would look very different indeed.