That mission pretty much begins with Shepard telling Asari soldiers to trust in their superiors and not pull out, because she's going to find something that will save them all. And then she doesn't. Instead she gets to listen to those soldiers scream while their world burns. Worse, she's now confronted with the fact that her failure leaves them with no leads whatsoever. They have no way of completing the device, no way of locating the catalyst, no way of fighting the reapers. She has to personally inform the Asari council member that her homeworld is destroyed, and then she has to tell Hackett that they might have just lost the war. So yeah, it's a significant moment.
But at that point in time, they don't know if Kai Leng destroyed the information. They don't know that all he did was stretch the timeline. All Shepard knows is that this was the one lead they had on how to complete the one device that they were putting all their hopes in. And it was snatched away.
And if you think Shepard thought they were going to be eradicated anyway, then I think you missed a lot of motivational speeches throughout that game.
Shepard just saw the Reapers obliterate all of Alliance's military and outright start conducting genocides on Earth. Somehow, the lives of some random Asari soldiers count? More people died on Earth in slaughter ships in the time that it took for Shepard to land on Thessia and have that conversation. That's my problem with these lives having any sort of meaningful impact.
As for there being "no leads whatsoever", that is literally the same position Shepard has been in every time the galaxy in danger. This is Shepard's job description by that point. As for telling the Asari their world is a crater, so what? Shepard just lived through his world being a crater. I don't see how this moment, in particular, is a big deal.
And when I say they were going to be eradicated anyway, I don't mean that Shepard thinks the war is hopeless. But conventional victory is clearly and completely farcial. There's one winning longshot here: the Crucible is a magic device that kills all the reapers. The catalyst is somehow related to this magic device that kills all the reapers, and Shepard screwed up.
But the whole problem with this "almost lost the war" is that it requires rationalizing how certain the incredible long-shot of the Crucible actually is in-game. Because for all Shepard knows, that things an auto-genocide switch like the Halo in the titular game. It's a weird way of treating this MacGuffin, which challenges the way every other conflict in the game already went.
At any rate: even if you disagree with me on all this and think I'm wrong, it's still my (and my characters) interpretation. Feeling stress is subjective. To me, Shepard would have been a wreck right after Earth, and on Earth. After that, it's time to get down to business: everything dies unless you can save the universe, and whining won't get you there. My way of dealing with stress is just so different from this that not only would I not react this way (my freak outs coming much earlier in the cycle), but it would never even occur to me that this is a time to be stressed out.