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.