The time-heals-itself concept is starting to grow on me
Sure, maybe Sera was an unreliable narrator, and it's just the USS Relativity working in the background to correct things (even if they end up happening a few years off from when they should).
But maybe in the time scale of millennia, it more or less evens out.
Khan comes to power later in the timeline, but DIS and SNW seem a little more modern than they should? Some things are slowed down, and others are sped up.
Why did they have to show Khan shifting into the future just one year after Picard and co. travelled back to the same time and we saw Soong on trial for the Khan project (a past event)? What benefit is there to jacking around with what they just established?
I don't think that changes anything that was previous established. Picard established that Soong did illegal genetic experiments, but it does not stipulate that his experiments were illegal because of a prior genetic research ban. Soong later recovers the 1996 Khan Project file.
My take is that Romulan time agents went back in time to hide the Khan file. They perhaps moved Soong away from that avenue of research not knowing how the Confederation would play into it. In any case it didn't work because Picard did his own time shenanigans pushing Soong to take more drastic measures.
In less than a year the Khan project has found a new home in Canada with new research partners. The Noonien-Singh Institute has Adam Soong's fingerprints on it somewhere, but it's unclear where. In any case whether there is a connection or not, Soong discovers the Khan project from 1996 in 2023. In SNW the Khan project has only just begun a year later and already has found its Khan.
It still seems like they could have coordinated the two plots in a more transparent way, given that the shows are running concurrently and have overlapping staff. Fans shouldn't have to do this much mental gymnastics to reconcile episodes that aired two years apart. The in-universe claim that the pre-history of our era is constantly shifting seems like a cop-out in those circumstances.
I take your point about wanting a more cohesive narrative, but I think there is a more important function served by the idea of shifting timelines. By allowing for the same essential historical events to occur just in different years, Star Trek can preserve what I think is one of its essential conceits: that it depicts our future.
I think the world of Star Trek was and is meant to be understood as a view into how we could develop, as a goal that we could achieve. Certainly, as a kid, that was why I found it so compelling. It showed me the great things that humanity could achieve if we decided to listen to and trust one another. It showed (admittedly not always very well!) that everyone has a place in the future, even people who are might currently feel hopeless, left out, or oppressed. While I can only speak for myself, I never felt that sense of purpose from other major sci-fi or fantasy stories. I may enjoy Star Wars or LotR, for example, but they don't mean as much to me because I don't feel like I or the humanity I know have a place in those worlds. They depict the dead past of a distant place rather than a living future that we could all have a hand in shaping.
I say that knowing that Star Trek is essentially fantasy, of course. My point is that, my maintaining the illusion that we are living in Star Trek's actual past, it makes us feel connected and invested in a way that is different from how we might connect with other stories. I don't know if that's the reason for introducing the concept of shifting timelines, but I think it still makes it worthwhile just the same.
I agree that ideally they would maintain that kind of fuzzy timeline to maintain our connection to their future. In fact, many years ago on the old Daystrom I tried to argue that we shouldn't take dates on the show literally other than as an indication of the general order in which things happened -- leading to massive pushback from almost everybody! It seems like Picard season 2 went pretty far out of its way to endorse the fan-favored theory that the Trek timeline forked sometime prior to the 90s, though, and I worry about the slipshod continuity management that is emerging as the streaming era matures. Of course, the Picard finale also abruptly undid the whole climax of season 2, so maybe the official position is that we're going to pretend season 2 never happened.
that we shouldn’t take dates on the show literally other than as an indication of the general order in which things happened
I absolutely am fine with this take now, especially. Ultimately, what we are talking about is the narrative importance of the events. If we want to have episodes where the characters go to "today" today must continue to exist within the narrative. If the narrative says that the world was noticeably different in 1996 - move it to 2036. If we're still going strong move it to 2056.
These changes don't make as much difference in the narrative that was being told to begin with.