They call him Zephram Cochrine of Alpha Cintari. The RPG said the same , he is Cintari. The Writers manuel said the same. But do the New Trek Writers keep to it , nooo , because humans is smart.
Humans rely only on themselves , so of course they invent warp drive independantly.
Secondly , Z Cockrine Supposedly vanished some 80 years before Kirk , who is at most 100 years before picard , placing him roughly in 2180 or so. 2160 at most. And now to the Borge. You couldnt kill the Borge off unless you killed ALL of them. Kill the queen , kill the borge.