"I read somewhere - perhaps on FR? - that the Turkish government (can this be correct?) - has legislated that no one can become Ecumenical Patriarch unless he was born in Turkey????"
As a practical matter, yes. The law now is that to be elected Patriarch, one must hold Turkish citizenship, but that's not something that just happens. +Athenagoras was the last non-Turkish born Patriarch. The EP has asked the Turkish government to allow any Orthodox clergyman to be elected and then give him Turkish citizenship as a dual citizen, but Turkey refuses.
This is the same great friend of ours, Turkey, that does not allow Kurds to name their children with Kurdish names, and does not allow Kurdish language in schools, yet they just can't understand why European Union just doesn't let them in not yet but close)!
BTW, once Turkey is in, tthe Turks can settle anywhere in member countries without a visa or permit, open up business and bring their entire harem with them. Accepting Turkey into the EU will be undoing centuries of Christian work of keeping significant numbers of Mohammedans out of Europe.
Norway brought 500 Pakistani refugees from Bangladesh (when it was East Pakistan) and today they have thousands. Sweden now has Muslim-majority districts. Islam will win Europe without a single bullet (of their own). The Europeans will use their own on themselves.