This is the most common fandom interpretation: that Sasha is superior to Jon, more qualified, and more connected, and that Jonah selected Jon instead of her because he was afraid of her and knew his plans wouldn’t work without her. It’s what you’ll come across more in fanworks, so I don’t blame you for coming to those conclusions.
However, it is 100% contradicted by canon.
Jonah flat-out says the reason he chose Jon in TMA 160; we don’t even need to guess:
I’ll admit, my options were somewhat limited, but my god, when you came to me already marked by the Web, I knew it had to be you. I even held out some small hope you had been sent by the Spider as some sort of implicit blessing on the whole project, and, do you know what, I think it was.
Jonah was looking for a trauma punch card, and Jon came pre-marked with the most difficult one to get. It’s really just as simple as that. Sasha didn’t have any marks that we’re aware of pre-season 1, though given that she had collected THREE before she died, it’s not a stretch to say she would have collected more just as fast as Jon did–or, given that she’s even more reckless than him (see: her running into the worms to tackle Tim in TMA 39), even faster.
But I also want to address this idea that Sasha was superior to Jon in every way, and the tragedy of the story would have been avoided if she had been chosen as the trauma punch card instead. People take Gertrude’s attempt at predicting Sasha as who Jonah will choose next as some sort of “picking her successor” or endorsement. She didn’t have any say! She was expecting Jonah to kill her, and was trying to guess his next move for who he would pick next. So instead of guessing who the trauma punch card would be, she guessed based off of who already had experience with knowing the fears were real: someone with experience in Artefact Storage. Gertrude and Sasha barely met, so all the deep insight and mentor/protégé interpretations that are popular don’t really fall into this. Gertrude was also barely aware of who Jon was.
So, there really isn’t a “Why didn’t Jonah pick Sasha?” because Sasha wasn’t even on Jonah’s radar for consideration, and there isn’t a “why didn’t Gertrude guess Jon?” because Jon wasn’t even on Gertrude’s radar for predicting Jonah’s next move. The whole Sasha vs. Jon thing is 100% fandom-made.
(It’s also 100% not due to sexism; that was supposed to be dramatic irony for our sakes, when we already knew Jonah’s reason since the end of season 4. Tim was just being Overly Aggressive Ally, but more than a bit misguided, given that Sasha was almost as unqualified as Jon was–none of the assistants had degrees in Library Science, Sasha included.)
For your speculation on how close she was to everyone, I would disagree there as well. In the Q&A’s, they talked about how all of the Archives employees are isolated and don’t have close connections with others–it’s one of Jonah’s prerequisites to transferring people there. Dozens of Season 1 Archives Found Family Sleepover Parties has kind of replaced this in fandom memory, but we don’t have any evidence that Sasha would have been impossible to isolate:
She knew Martin was lying on his CV, but not from him. She hacked into his files for the fun of it and then gossiped about it to Tim. If they had a very close and trusting relationship, I wouldn’t see why she’d resort to subterfuge, and I think she’d know how desperate he was to keep it secret.
We don’t see them interact in season 1, and Sasha seems to have a low opinion of Martin’s competence: she thought both Martin and Jon were overreacting to the danger of Prentiss AFTER Martin started living in the Archives. She thought it was silly that Jon was taking Martin so seriously, because if Prentiss was really a threat, there’s no way someone like Martin could have survived her, right? Frankly, even though she’s less snappish with Martin than Jon was, it’s very fair to say she had a lower opinion of him than even s1 Jon did.
She was friends with Tim, but I think even that gets exaggerated in fandom. She loves to gather people’s secrets, but shares none of her own: for instance, Tim told her about Danny, but when Sasha started having encounters with Michael, she kept it all secret from Tim. She said she didn’t want to tell Jon without more evidence, but what was keeping her from giving Tim a call before she ran off to meet a monster by herself in a graveyard? I feel that if she were Archivist, this would have been even more intensified: she would care about him, sure, but that’s nothing new: Jon cared deeply about Sasha and was devastated by her death, but that didn’t prevent him from being isolated.
And I don’t need to put too much here for why she’s just as bad as Jon, if not worse, for needing to solve mysteries and rush into danger for information. Just listen to her statement. Jon spent all of season 1 hiding from the truth, while Sasha met with a creature she knew wasn’t human multiple times, by herself, without telling any of the others, all out of curiosity. Jon started invading people’s privacy in season 2 because he was being driven to paranoia by the presence of the Not!Them, but Sasha was already invading all her coworker’s privacy in season 1…just for fun. She was VERY suited to the Beholding, arguably more than Jon was, and would have been very easy to manipulate throwing herself into mysteries to get more marks.
Frankly, because she only appears in 5 episodes, fandom Sasha ends up being about 99% OC and a bit of a wish fulfillment character; most examples people give for why she’d be better at Jon at everything, find out all of Jonah’s secrets, avoid losing any assistants, and save the world is just….based off of headcanons, vs. the scraps of information we get from the actual episodes.
The reason why I dig so deeply into all of this and care so much about this interpretation is it changes the entire nature of the show if oversimplified. If Sasha really is this ultra-superior being who would have fixed everything with her mere presence, the tragedy gets whittled down to a pretty uncomplicated “everything would have been fine if this Hypercompetent Woman was in place instead of this loser man.” But that’s not the story being told. The tragedy we have is that whoever was doomed with the Archivist position would have been lied to, manipulated, tormented, afraid, isolated, hurt and hurt and hurt again. If Sasha had been chosen, she would have been the one suffering through all that instead of Jon, and that’s the narrow miss that we’re left to realize from Gertrude’s tape.