The issue existed prior to Michelle Paradise. Season 1 had two different world-ending scenarios (the throwaway line about all life in the multiverse dying due to the ISS ship, and the Klingon threat to the Federation). Season 2's Control arc almost certainly predated Michelle Paradise having a major say as well. And let's not forget that every single season of Picard also featured an existential crisis of a similar level. This is on Kurtzman, not her, I think.
In the end, I think the issue is threefold:
1. Modern Trek has (much to my chagrin) stuck to plot-based writing rather than character based writing. The best stakes in any story are personal, but when interpersonal conflict is limited, it's hard to really leverage those personal stakes.
2. There's no sense of long-term planning from season to season. Whereas something like DS9 had semi-serialization and a long-term plan regarding the Dominion War, Discovery seems to have never planted the seeds for a multi-season arc, meaning it's not building towards anything, but more of the same.
3. As a story with galactic-level scope, setting the stakes of an entire season as something smaller (survival of the ship, saving a colony planet, etc.) can feel underwhelming.
So it's just wash-rinse repeat. And as a result, there's a bit of a "boy who cried wolf" issue across Trek series now.