Who says German e is supposed to sound like an English a? The a in "Nate" is completely different from the German "e".
There are actually three forms of the German e: the short one (which is closed), the long one (which is open), and the final one (called "schwa" in English). "Net" or "get" is a good English representation of the short German e.
The final e (e.g. "habe") sounds a bit like a very short English a (and not like the long diphtong a in Nate), but not quite. Maybe that's where the confusion comes from?