Interesting analysis, but there is another time were Sakura's sympathy cracks : when Misato arms the DSS Choker and Shinji asks for it to be removed, we see her looking up and murmur that "there's no way that's ever coming off." That was her exact word according to the GitHub translation.
So even if she sympathize with Shinji's situation, she sill thinks that he deserves the punishment of the DSS Choker, which is I recall basically a fancy bomb collar for Angels!
As for the reactions of Midori and Sakura, another possibility is that they are annoyed by his request because they don't find he's worthy of helping(and even less piloting EVA-01) : remember that by the time of 3.0 Shinji is the only child left in the Wunder, everyone else is an adult, even Sakura and technically the two pilots. So Shinji is not only a kid, but also proved on more than one occasion that he's unreliable(when he lost it against Shamshell, when we attacked the HQ after Bardiel, the fight against Zeruel where he accidentally triggered N3I)... and now you have the same emotive and unstable kid who didn't changed at all since the past 14 years since he was "frozen", who comes and request to pilot again to help them!
You probably saw various series of movies where you have a group or crew of military/adventurers/explorers highly trained and all adults, and with them there is a teenager who want to join the fight? Usually he's portrayed as an annoying kid who don't know what he's doing and is shooed away by the crew who told him this is "adult's business and not a game".
Well here the crew is the Bridge Bunnies, and the annoying teen is Shinji.
In that shot of Midori's reaction, she's also letting a "tch" of annoyance, like the one of an adult irritated by something stupid a child just said.
Another possibility is that it's not really the fact that he want to help that touched a raw nerve, but the fact that it's EVA-01 he's talking about : for all we know last time the older ones saw it, it was becoming a god and launching an Impact, and the FoIs that must have been the most insane and nightmarish part of Third Impact were all copies of EVA-01, suggesting that she had something t do with these dudes. It's very probable that EVA-01 became the image of the devil in the collective mind(the JSSDF Commander already called it "the devil" in EoE), when someone think of the face of evil and destruction, EVA-01 comes in mind.
And now WILLE have to use this
thing to power their battleship, it's possible that they previously tried every other options to power it before resigning themselves to free it from its space prison and put it in the heart of the "Ship of hope", so they are understandably less than thrilled at the prospect of letting it loose, especially if the one asking for it is the pilot of this thing!
Also, it could be that they don't know if he's human, or that a wariness against everything related to the Evangelions, pilot included, has grown with the years, and Shinji is suffering from this discrimination. After all the Evangelions and Angels are the responsible of all the suffering mankind has endured since 2I, and currently the known pilots consist of an half-crazy British girl, a German fury who have her right eye
glowing blue on occasions beneath her eyepatch, a blue-haired red-eyed girl continuously cloned to keep fighting against them, a fucking
Angel(if Kaworu's existence is common knowledge, since the DSS Choker was built for him I suppose it's the case), and a boy who passed the past decade and half fused with the "Devil"! And none of these guys seems to age at all since the past 14 years!
So sure Sakura is sympathetic toward Shinji for everything he's enduring(Toji probably told her how much he was suffering in trying to protect mankind, notably the time he was in Shinji's entry plug when he fought Shamshell), but still thing that he deserves at least the DSS Choker as punishment. Now the question is for
what exactly he's being punished.
Oh, and about Sakura's line in Japanese that SDD posted on the previous page :
ホンマ、カンベンして欲しいわ
Most of the online translators give me : "I want you to forgive Honma", or the "In truth, forgiveness want to."
But, Google translate give me : "Really, I want a break"!
So... which one is correct, because that line have some influence in Sakura's portrayal?