This is a really ambitious game. The cooperative storytelling aspect requires all players, and none, to simultaneously act-in as the GM. The story revolves around The Team and The Commander, the former doing the latter’s dirty work. Who, where from, what is the Commander? These are the kinds of questions this game will have your table asking, between missions where you fight aliens and save humanity.

For this concept to be easily realized at the tabletop, I see two prerequisite developments -

1. A conflict-resolution mechanism for players to mutually control the Commander. Perhaps each player represents a faction who has a stake in the Commander’s choices - e.g. his family, the civilian government, the alien parasite slowly growing in his brain, perhaps? At present, it’s not clear what motivation the players have to pull the Commander in opposite directions, nor what internal glue is keeping him together.
2. The creation of some kind of artifact. This is just my opinion, but I find that cooperative storytelling games work best when you create something tangible together - be it a map, journal, etc. It doesn’t even have to be permanent. But creating something together, beyond just words, helps moderate and cement the activity.

All in all, an interesting concept that is well on its way to being a complete game.

I think this is a great start - and a good place to start playtesting. Something that I think might be interesting is to include a way for cumulative Stress/Trauma to influence the difficulty of later actions. This could be along the lines of "adversarial dice" that get included, or even changing the number of successes needed per objective in a complex roll. 

I love the idea of each player just taking a chunk of the cast and working together to make a story, I used a similar mechanic this jam. I have only a passing familiarity with Evangelion and your other influences but I could still recognize some of the tropes. With a little polish, this could definitely appeal to fans of those titles, congratulations on finishing this version!

There’s some great stuff in here – I love the Handler/Commander being player roles rather than NPCs. Your system of Pilots going off-mission, Handlers aborting, and Commanders revealing secret objectives is fascinating (I forgot how much mistrustful and secretive adult-kid dynamics drove Evangelion’s character drama)!

Would love to one day see a version with less text-heavy layout or with your examples laid out in a template mission. You might like SAD TEEN MECHA PILOTS as another directly Evangelion-inspired ttrpg!