I have no particular stance on whether estrogen would have saved Shinji, but I'm looking at it from the family dynamic angle, and... well, Jesus Christ. On a good day, Shinji already looks like a teenage soft butch version of Yui Ikari – can you imagine how Gendo would react to his "son" slowly morphing into his dead wife?
Asuka remarks that "y'know, now that the hormones are starting to kick in, you look an awful lot like Rei" and from there, accidentally unravels the entire plot within minutes.
