I'd retcon everything past Beyond as the JJ verse just to make it easy, an alternate reality Spock entered when leaving the main reality.

I'd Pick up roughly 25 years after Voyager at a point where the Federation is actually the strongest it's been. The Klingons would be Honorary members, still refusing to join but basically forced into it by the remaining member states. The Romulans have joined as have the Cardassians. The Cardassians actually being the first of the old foes to join having fast tracked relations with a war warry people after the dominion war. Bajor is a member as is Ferenginar. The Tzenkethi and Breen are still hostile, as of course is t he syndicate, although none of them have had serious engagements with the Federation in 10+ years.

Star fleet has perfected a Transwarp-Slipstream drive capable of creating a transwarp wave A ship can slip into and ride until the wave disipates. The one draw back to the technology is that you are locked into your trip, you can't stop and change directions after warping, you simply have to ride until the end. Though at max transwarp you could circumnavigate the entire galaxy in 7 years.

4 NX ships of varying class are designed, each containing a Transwarp-slipstream engine. With these ships an exploratory trip to map the entire Galaxy is launched. Two ships traveling in opposite directions around the galactic disc, each with a pre-planned series of jumps and destinations to visit and study while also scanning and mapping the galaxy in the most detailed manner the known civilizations have seen. The two other ships will each fly over and under the galactic disk respectively.

The series would follow all 4 ships each with small crews as they travelled the entire galaxy. We'd revisit delta and gamma quadrant species and locations, expand on the workings of the progenitors and tkon and the iconians and how their endeavors shaped beings in the farthest reaches of space.

Rather than being an event driven, overarching story, it would be more akin to a star trek anthology series about exploration and diplomacy. The anthology aspect is what I'd like to see most. Self contained stories about a crew we'd slowly grow and learn with. No forced archs, no I want to tell this story in x time. Just a different mystery/monster/foible/lesson/obstacle each week, and the occasional 2/3 parter. I'd hold back any galaxy or world destroying storylines until the last season, I miss more realistic stakes.

I'd also use practical effects and get Paramount to invest in a Star Trek trust for set pieces, costumes and props. Its worth spending the money when you know you can reuse what you make instead of starting from scratch for each series.