With Picard, Alex Kurtzman is really picking up someone else's Star Trek -- Rick Berman's -- and continuing with it. With the third season of Discovery, he has a chance to truly develop a Star Trek universe of his own. One that isn't butting up against TOS or is continuing TNG/DS9/VOY.
In the world of the 32nd Century, I think they'll enjoy having a canvas that's completely their own and they'll want to keep building on top of that material. Picard and the 25th Century are a fail-safe in case the new setting in the 4th Millennium isn't well-received.
What I'm hoping is that the DVD/Blu-Ray Commentary for "Such Sweet Sorrow, Part II" will delve into what specifically led them to decide to switch timeframes and how far in advance it might've been thought out. I think it's earlier than we've suspected but can't say for sure. Even though I got the first inkling they might do a bigger, more serious time-jump later on after they did a small one after the Mirror Universe episodes, I think they might've been laying down groundwork as far back as "Magic To Make the Sanest Man Go Mad" by introducing the concept of the time crystals. Even if they didn't know what exactly they wanted to do with them yet, but they went out of their way to put them out there in the most unsuspecting way. I don't think it was random.
If they want a series about a Federation that's fallen from grace, if not fallen entirely, then they probably wanted a crew that actually remembers the Federation first-hand to help rebuild it. So in order to do that, it helps to show the crew actually living in a time when the Federation was around and in its prime first, learning from their mistakes so they can show people in The Future how it's done.
So I think at the end of Discovery's second season, they wrapped up the first stage of what they want DSC to be. In this next stage, in the permanent setting where the show has ended up where they wanted it to be, it might take some time to tell this new story. I certainly don't think the Federation can be rebuilt in two seasons. Andromeda might've rebuilt the Commonwealth in two seasons, but I think this would be a "rest of the series" arc for Discovery. Rebuilding the Federation seems like it should be a long-term goal. Not a short-term goal.
The showrunners may have changed, but Alex Kurtzman is still the one ultimately in charge and who the showrunners ultimately report to. So I think they had an overall idea in broad strokes, it's just the details that changed. Most prominently, the details probably changed with the shakeup after Gretchen Harberts and Aaron Berg were fired. I think the idea of Discovery ending up in The Future was always there, and remained intact, it's just that how it got there is what changed.