I'm not sure I can articulate what I want to say in answer to the OP's question, but I'll give it a whirl...
1. If I were in charge of "rebooting" the franchise, I'd take an approach that almost entirely pretends that Star Trek never happened. We never had the show we had, and we never had the 51 years of history, etc. The reason for this isn't just to "start fresh" or "clear up the baggage"...but it's to completely update the show. I'd then start right where Roddenberry and his team started back in 1965- let's make a smart science fiction show about exploring the galaxy. Take almost everything but the basic framework off the table. Which leads me to....
2. I'd retain some of the key elements. I'd have a USS Enterprise as the primary "vehicle" for the characters and plot. The design would be vaguely familiar (saucer, nacelles, etc) but would otherwise be very re-imagined. The characters would get a "NuBSG" upgrade...keeping somewhat familiar names and positions, but having gender / race etc be arbitrary. I'd keep the "logic, passion, reason" triad between the Kirk, Spock, and McCoy characters. I'd add a few new ones (love the idea of the female security chief).
3. The technology would be different. I might look at a different kind of FTL drive, based on whatever the most up-to-date science theories are. Someone said "transporters are really micro worm-holes or something" yep...that's a good example of an update I'd put in place. No holodecks (shared experience VR perhaps). Tricorders would have to be something new and amazing like an iPhone 900 or something. Phasers and communicators can stay.
4. The visual style would be more like Interstellar or Gravity, especially in planet orbit. I'd really aim for the most realistic portrayal of "ships in space" that could be achieved on a given budget. No sound. No singular plane. Make it visually unique
5. No United Federation of Planets as we know it. Perhaps it's a loose association of member worlds who are aligned for trade, technological exchange and mutual protection...but otherwise that's not a central focus on the show. This would NOT be a "world-building" show at all. Which leads me to...
6. No intricate tapestry of alien governments and superpowers across the quadrant. This show would be about an exploration cruiser out on the frontier...far from involvement of "home base" and all the macro-complications that come with it. Aliens in this version of Trek are still very "alien"...especially non-member races. Klingons and Romulans, for example, could be completely re-imagined (COMPLETELY) as mysterious, hostile beings that we know very little about and appear sparingly. Encounters with them would be more like "Balance of Terror" or "Arena" with the Gorn rather than "Errand of Mercy." There is mystery around these races, they are unknowns. They are very alien. Maybe we don't even yet know how to communicate. Their technology would be very different (instead of basically the same as our tech but with different aesthetics).
7. The show would take place in a period about 400 years after a "colonization boom era" sent sleeper ships into the far reaches of space to start new worlds (think Alien: Covenant or Passengers, for example). So now, the Enterprise is not just exploring "strange new worlds" but is also one of the first vessels in history to be able to investigate some of these colonies and societies and see what has developed in the several centuries that they've been up and running. This would give you the opportunity to do some of the "parallel worlds" stuff Star Trek seems to thrive on, and would also be supportive of why there are so many "human looking inhabitants" out there on these planets we discover. I think it's also a nice opportunity to say something about humanity...what did these pioneers come up with? What type of society or life did they build? What are their current challenges?
8. It is a human-centric show (sorry everyone who's clamoring for "TEH MOR ALIENZ IN TEH CROO!!!1!") Star Trek's heart and soul has always been about humanity. With my thought about de-centralizing the UFP...Starfleet now becomes Earth's exploration and defense force. Each UFP member world has their own such organization which functions very differently. There are exchange programs (which is how we will get Vulcan Spock aboard in this show) similar to how we got Phlox in ENT, but we are not looking to have a mixed crew. This show will be about humanity's growth and struggles out amongst the stars. Aliens will still be very "ALIEN" to us in this show. Even UFP member worlds are still largely mysterious and filled with unknowns. I think that this would lend itself to greater realism and greater practicality, as well as improving one of the MAJOR flaws in Star Trek as I see it: Aliens should still be very much unknown wonders to us... that's part of the excitement and wonder of this show.
9. Going back to the "on the frontier" thought...this wouldn't be a "return to Earth every 4 episodes" series. In fact, I'd bring back the TOS rule that Earth is rarely, if ever seen or even heard from. Our pioneers are out on the frontier for 5 years with minimal to no contact with home. Much of the drama comes from this community of people who really only have each other for such a long period, and are making decisions and executing on things on their own authority based on their training. Subspace transmissions must be specially set up in person at positioned relays throughout the galaxy, using some technobabble technology. You can't just call up Admiral Humperdink on the view screen to get instructions.
10. No utopian post-scarcity bullshit. The "optimism for the future" element so key to Star Trek is reflected in the fact that humanity is no longer killing and waring with itself and people of all races, genders, and belief systems (religious, political, etc) have learned to co-exist and work together DESPITE their differences. I think, especially in this world's current climate of polarized opposition in every area of our lives, THIS would be the key point to show people that we can work together. Humans are still as flawed and struggle as much as always...we just don't act like prejudiced hateful assholes toward each other because of different values and beliefs. But, there are still struggles (economic, resource, social, etc) that humanity has to overcome.
The following elements, by letter of law, would never appear on the show:
- The Borg
- Time Travel
- Reset Buttons
- Politics in SPAAAAACE
- War (action and conflict---absolutely! But more like "Arena" where it's a 1-on-1 fight with the unknown)
- Section 31
- Holodeck episodes
- Evil Admirals
- Speeches about humanity's superiority
- particles or deflector dishes as solutions