This is an idea I came up with in the "Yui's an evil bitch" thread, which can basically be summed up as, "What if Evangelion was a played-straight shonen mecha anime?"
The story background is the same: Angels attack, Seele instrumentality plan, Shinji gets called in because he's uniquely qualified to pilot Unit-01 for reasons. However, the point of divergence from Evangelion is that Evas are made from soul imprints, which means that Yui and Asuka's mom are both still alive. This, naturally, has some pretty far-reaching implications on the other characters, which can be mostly summed up as "everybody isn't nearly as fucked-up as they are in the show".
Gendo undergoes the biggest change: instead of being a crazy person who wants to bring his wife back from the dead, now his primary motivation is to ensure a safe future for his family by eliminating the threat of the angels and sabotaging Seele's plans for instrumentality, by any means necessary. From Shinji's perspective, he'll still seem somewhat cold and distant, especially when he sends Shinji off to some boarding school far away from Tokyo-3, but he really just doesn't want Shinji anywhere near this conflict and a lot of his apparent coldness just stems from a lack of social skills and an inability to more closely relate to/connect with his son. While he might have trouble expressing it sometimes, there will be zero doubt that he loves his family and will do anything for them.
Yui will be the one who calls for Shinji to pilot, because she knows that he's the only one that can do it (for reasons that will be elaborated on later). She's more willing to put him in harm's way (as opposed to Gendo who doesn't want Shinji anywhere NEAR the Eva program) but it's because she believes that her son has what it takes to save the world. Where Gendo is over-protective of Shinji almost to a fault, Yui might seem a little TOO eager to get him in the pilot seat.
Shinji will be more like his manga counterpart. He's still kinda shy and soft-spoken, but he's generally just more of a normal teenager with normal teenager problems and normal degrees of teen angst, especially regarding his cold and distant father who Shinji thinks doesn't love him, because why else would you ship your kid to a boarding school on the other side of the country when they're like 8 years old? This is intensified once he gets to Tokyo-3 and sees his parents basically treating Rei like an adopted daughter while he got sent way. But yeah, main gist is he has a similar personality but isn't severely emotionally crippled and is generally able to interact with others in a healthy manner.
Asuka, like Shinji, will mostly keep her personality, just toned down significantly. Her mom is a workaholic who rarely has time for her, so she retains some of her attention-seeking behavior. Like in NGE, she gets selected to be a pilot at a young age and has been in training for the longest.
Rei I would be re-written to satisfy some plot requirements. The Ayanami series was nominally created (like in the show, from a hybrid of Yui and Lilith's DNA) as an experiment in creating artificial Eva pilots, but are really a contingency plan by Gendo & Yui to act as their own impact trigger if they're unable to stop Seele any other way. Or... something of that sort, the why isn't that important. Rei I is the first, and would have appeared to be around 16-18 years old when Shinji was a child. She would've lived with the Ikari's, and basically would've acted like Shinji's older sister. And this kinda leads into the next big change:
Unit-01: Contains Rei I's real soul (Unit-00, which she tested with initially, contains an imprint of her own soul). Because Unit-01 and Rei I were both grown from Lilith, an unforeseen reaction occurred during a contact experiment with Unit-01 which resulted in her getting absorbed into Unit-01. Shinji was there to witness it, but he either buried these memories or they were erased by NERV. It's immediately after this event that he gets shipped out of Tokyo-3. Unit-01 is thus special in being the only Eva unit containing a real soul, with Shinji being the only person she accepts as her pilot.
Rei II is, like the other kids, mostly keeping her general personality. She's quiet and bookish, and is mostly disinterested and disconnected with others because she's spent most of her life in the NERV lab and is out of school a lot. Gendo feels bad about this and goes out of his way to act like a father to her when he can. Seeing this, Shinji gets a little jealous because he feels like Gendo never showed him any of that same affection.
Any characters not mentioned are mostly unaffected by Yui and Asuka's mom being alive. This presumably means that all of Class 2A's moms are also alive, though the implications of this will not be explored as their evident lack of moms was never really a major influence in the show anyways.
The story starts similarly. Seele is anticipating the third angel will attack soon, so Rei starts more training with Unit-00, with one of these sessions resulting in an accident that leaves her injured. With just days before the projected angel attack, with no time for other NERV branches to ship over their pilots and Evangelions, Gendo is left with no option but to take Unit-01 out of ice and to call Shinji to Tokyo-3. Misato will still be Misato; she picks up Shinji when he arrives in Tokyo-3, and upon learning that Gendo and Yui were working so much that they were practically living at NERV HQ and really had no time to take care of him, volunteers to have Shinji live with her during his stay in Tokyo-3. Events from there will play out in a similar manner, except instead of everything going to shit because the characters are unable to properly communicate with each other because they're all horribly emotionally crippled, they're instead able to support each other through their problems and help each other grow into better people. While they're able to defeat all the angels, despite Gendo's best efforts Seele still manages to pull through with most of their Third Impact plan. Rei chooses to sacrifice herself in order to save her friends and family by hijacking Seele's plan and triggering 3i herself, giving Shinji the reins. Within Instrumentality, Shinji sees an alternate universe where his mom had died during a contact experiment and how that really seemed to fuck everything up for everyone. Thus, he realizes that no matter what difficulties he may have dealing with his friends and family, it's NOTHING compared to the possible future he saw, and that he really ought to appreciate the people around him who truly do care for him, even if they sometimes have trouble expressing it, and decides that he wants to return everyone to the real world.
I doubt I'll ever write anything for this, but it was fun thinking through a what-if scenario of "Evangelion as a typical shonen anime".