If it was up to me
(it obviously never will be), new Trek would:
1. Take place in the old continuity.
- there's a lot more background to draw from.
- personal preference.
2. Keep being relaxed about "proper military procedure".
As far as I'm concerned, people that agonise over a poor "at ease" stance or whatever are completely missing the point of Trek.
3. Resolve canonicity on a case-per-case basis: maybe ToS is authoritative on X; but TNG/DS9/VOY on Y and ENT on Z, depending on many factors; but we have to be willing to accept some things won't make sense in the past series.
Example: most "maps" of the galaxy place most of the Federation, most of the Klingon Empire and all the Romulan Empire in the Beta quadrant despite most series (especially DS9&VOY) calling them "powers of alpha quadrant" or talking of "the fate of the alpha quadrant" or "going back to the alpha quadrant"... the new series would use completely different maps.
Once canonicity has been decided, stick to it for the whole run.
4. Now for the interesting stuff: focus on the Federation/alpha quadrant as a whole, not a specific Ship.
Instead of having a regular cast of immortal (plot armour) idealised heroes, the series would span a lot of characters, some dying quickly, some behaving in a less-than-perfect fashion, some manage to beat the odds once but become background characters afterwards, some perform vital but understated roles,... you can have many potential stories in such a setting, be they one-shots about a "back office" character or geopolitics or true exploration of a new sector or resolving a tense situation through diplomacy (or failing to!)...
Such an approach can work and can be successful, as GoT showed (though this would go further).
Pro: Many clichés avoided, stronger storytelling.
Con: Viewers don't get as attached to characters as they otherwise would.
5. Build an actual (abstract) galaxy.
What do I mean? Simple: decide early on what star system is where (in relation to X reference point); where FED/KDF/ROM/other Ships are in the galaxy, what's their mission and capacities, who's their commanding officer and so on.
Not all of that info has to appear onscreen; but it would help in having a consistent world.
No Deus Ex Machinae of X Ship saving the heroes on (at?) the nick of time, you know in advance if a Ship is within range or not; no sudden appearance of a full hostile fleet unless it was pre-planned, same for random nebulae... basically, this approach constricts the writers somewhat, but once again, it prevents a lot of clichés.
Especially if geopolitics/war is important to the setting.
There could still be spontaneous elements, so long as they made sense within the overal setting.
Will it happen? No. Still nice to theorise.
So, what'd you think 'bout my "ambitious but rubbish