I wish this show wouldn't challenge my suspension of disbelief so... violently.
Yeah, a ship from early TOS era that's not designed to go through atmospheres is able to bust through solid blocks of rock, tear through an atmosphere and crash land on a planet with only a few scratches that take a few hours to fix. At least mention how incredible that is -- throw in some nonsense mumbo-jumbo about how it was able to do that because of some unlikely thing or another.
I think that they did First they talk about doing a thermal roll to let the shields take some of the brunt. Then Saru orders the graviton beam to be readied and to use the deflector shields in burst to cushiion their landing. They fire what is presumably the graviton beam. One of them says the shields are off-line. But they survive. Then the bridge crew erupts in applause and congrats Detmer.
So plenty of nonsense mumbo-jumbo, and if it makes you feel better, feel free to fanwank that the crew is saying, "Holy shit, we were lucky to pull that one out of our ass!"
But no - no one even asks if the ship is able to fly. It's just taken as a given that it'll be able to.
In the beginning of the episode, Saru asks for status reports. The crew gives them. There is apparent damage to some of the internal systems. Reno literally says because of that damage, "We can't fly." Now you can either take it as a given that early on they were able to assess that there was no structural or other problem that would prevent the ship from launching, or that they decided to fix the known problem that would prevent the ship from flying and then do an assessment later to see if there were other problems.
At the end of the episode before the attempted liftoff, the crew updates Saru on stuff again. No one mentions anything suggesting structural damage. Saru talks about how starships weren't designed to lift off like they are about to do but they have no choice. So again, you can either take it as a given that Discovery didn't sustain any structural damage that prevented it from being able to fly, or that in the time that they had with all able-bodied crew and the robtots constantly working, they were able to fix whatever structural damage there was.
Tilly still behaves like a painfully shy, verbally incomprehensible first year academy student. It was cute at the beginning of the first season, but at this point, given everything she's been through, she should not come across like someone who constantly needs to be tucked in bed with a pacifier.
Two years(?) of experiences are not necessarily going to change entirely the character one spent decades creating. This seems like a Catch-22 to me. If she had acted braver and bolder, it would seem inconsistent with her established character.
Similarly, Stamets still behaves like a petulant child in his own ways and has seen little character growth. There was no reason for him to try to make that repair on his own - at least none that was presented to the audience. A reasonable person in his condition would've at least attempted to recruit an able body to assist. Show us that that wasn't possible.
I think part of the point is that Stamets is often a control freak to the point of being an unreasonable person. From injecting himself with Tardigrade DNA to wanting to jump back to duty when he couldn't even spell the sentence his husband gave him, he aggressively wants to do what he thinks is necessary for the good of the ship, even if that thing is dangerous and illogical.
Even having been told that Saru ordered everyone to work in pairs and knowing that he just barely got on his feet, Stamets says he can do the job on his own. Jett expressly talks about Nilson, another crewmember who could do the task, and Stamets says no and insists on doing it himself. Hopefully with the benefit of hindsight, Stamets sees that his stubbornness was a bad idea. Or not. Neither people in real life nor characters always digest the lessons that experience should teach them.
Detmer has suffered some head injury and is stumbling around oddly, suggesting that something particularly unusual is going on with her, but nothing comes of it. She just stumbles oddly throughout the episode with no further enlightenment on her situation.
There are a lot of crew to follow. By nature of the format, it could be something that takes a couple episodes to pay off or it could be that they've dropped it. Too soon to say what to make of it.
In the previous episode, Michael practically begs a stranger for help within hours (minutes?) of landing on the planet. I can't imagine any Starfleet officer on any other Star Trek show falling apart so immediately. Not that it wouldn't be understandable in certain conditions, but give the audience a reason to believe it. This is supposed to be an exceptional human being with exceptional training and experience. We're used to seeing Starfleet officers go through days of literal torture without showing that kind of frailty, but here Michael immediately asks to be taken in by someone she knows nothing about.
First, it is rare that we have seen a Starfleet officer in similar circumstances to Burnham's (fresh off learning that her deceased mother was alive and losing her again, seeing that her brother had been insane and accused of murder, facing an existential threat to humanity, thrust into an unfamiliar time and space, not knowing what happened to the rest of her ship,)
Second, I don't see anything she said or did as falling apart. Honestly, it seems to me that it is remarkable that she is as together as she is. Again, sort of a Catch-22: if she was more together, she would be labeled as a Mary Sue (more than she already is).
Third, in TOS alone, here's a partial list of Starfleet personnel who fell apart under various circumstances:
Galileo Seven: Basically everyone but Spock and Scotty freaked out at being marooned, at a death among the crew and facing the giant aliens of the week.
Star Trek 2009: Kirk when he is marooned on the planet that Spock Prime happens to be on is about as "falling apart" as Burnham is.
Omega Glory: Captain Tracy has gone full Col. Kurtz by the time we meet him
Patterns of Force: John Gill thought the best way to help the people of the planet was to institute Nazism.
The Ultimate Computer: Daystrom, despite his brilliance, has a mental breakdown.
Whom Gods Would Destroy: Garth, a highly decorated officer, is literally insane.
The only character with reasonable development and who's acting as one might expect a Starfleet officer to behave is Saru, so at least we have one character that doesn't constantly challenge my assumptions about what a Starfleet officer is.
Maybe you should re-evaluate your assumptions about what a Starfleet officer is or might be.
At the end of the episode, everyone acts like it's mind-bogglingly shocking that Michael has been waiting for Discovery for a year. But it's been established, and all these people know, that the time of their arrivals could be wildly different. Are they shocked that it was ONLY one year? Because they're acting like she just said "I arrived 20 years ago".
I really want to like this show, but the writing makes it such a challenge.
I think you are exaggerating a lot here.
First, I don't think the looks of the crew are necessarily that of being mind-boggled.that Michael was there for a year. There are probably a lot of things they are processing, including nearly being stuck on Hoth-light, the fear that they were dealing with hostile aliens and were mostly defenseless, and their relief at it actually not just being a friendly face but Michael herself.
Second, I don't know if the mechanics of time travel have been so established for the crew of the Discovery. We the audience know that time travel is arbitrary that going through it could mean that despite using the same method of travel, two objects can end up in completely different times/spaces. But we the audience know this because we have a number of in-universe examples of it. Where/when have the crew of the Discovery (excluding Michael) been given this knowledge?