brunos

This question has been asked in many threads.
I have never seen a definitive answer.
Caling the consulate could help, but I have found the consulate where I reside unwilling to commit a clear answer. Very risk averese.

As Often1 mentions. each country has its own rules. So getting into one country does not necessarily means you an get into another EU country..
In Timatic the wording is often variable depending on the country.
Apparently France allows both nationals of EU+ and pax "arriving" from EU+.
They also allow pax "arriving" from 13 new countries.
Germany allows nationals of EU+ and residents of 8 new countries.
The use of words "national", "resident", "arriving from" is confusing, and I am not aware of anyone providing a clear answer.

As to your question about residency versus nationality: If you hold say an Australian passport, you do not have to provide proof of residency there. It's only if you reside in Australia with another nationality that you would need proof of residency in Australia. In other words, nationality assumes residency for travel purposes even if that is not the case.