What's interesting is that regardless of when the flagship show is set, "present day" of the Star Trek Universe is (currently) the early 25th century. It's not as easy to say why this is (it's not the farthest future anymore) but it still holds true even with Discovery in the 31st century.
For my part I absolutely think the franchise should center on the 25th century. I think the 23rd century focus is an artifact of a misunderstanding about the nature of Trek and what brought fans into it from the 2000s that walked on for a decade like a zombie.
In my idea vision:
- the 23rd century, the past, would amp up the silly adventure to the nth degree, much like SNW seems to be doing in Season 2. This is where you go for action adventure. This is a saturday morning cartoon.
- The 25th century, the present, would focus on long running plot lines and characters, be more "realistic" and relatable and continue what Picard Season 3 did. This would be more talky, a bit slower, a bit more grown up.
-the 32nd century, the far future, is where the weird, experimental shit goes. This is where the franchise can take its biggest, most disruptive risks (can always cut off the future timeline). Wanna crash the moon into earth? Do it.
We basically have that, albeit the 25th century show doesn't exist yet besides Picard, which is over.
I think the 25th century is "the Present" for two reasons. First of all, the 24th century cast/crew/stories are the ones that the large majority of Trek fans grew up with at this point, and is the largest body of work. SO it's by far the most familiar.
The second reason, connected to that, is those seasons happened in near real time (1 season of the show = 1 year in the Star Trek universe), and even with the gap the resumption of 25th century stories happened in near real time. We're a few years off in actuality, but it's damn close to picking up where we left off in 2002 when Nemesis ended the 24th century run that began in 1987. This mattered most of all, because it has allowed characters and actors to age in real time. Voyager ended 22 years ago, and Jeri Ryan is 22 years older, and we're all 22 years older, and the story is set about 23 years after Voyager returned.
That's why it's the Present. Because we see ourselves in Stewart, Ryan, Frakes, McFadden and so forth. Yeah they're older, but so are we. This is why I think if they did a live action show set in, say, 2382, it wouldn't feel like "the present" but rather the past, which is why doing the post-Nemesis era in animation is clever. If they started doing a live action series and everyone was wearing First Contact uniforms still, it wouldn't feel current, especially not after Picard Season 3.