Set anywhere between the end of the Romulan War and WNMHGB, show the Federation and Starfleet in their younger days, hopefully with a lot less upper level politics (one of the things that killed Enterprise).
Need someone fresh in command, neither Kirk nor Archer, A more Steven Decatur like character, young - brash - headstrong - driven, exploring the space surrounding the infant Federation. He a veteran of the romulan war, this is his first command, given his youth he's likely only a Lt.Commander or Commander. His lack of "seasoning" result in his making the rare mistake.
The ship will not look like a corridor from a Hilton hotel. Make it tight inside, world war two submarine tight, people squeeze past each other. (Might make it difficult to get cameras into). The crew has families, just not aboard with them. There are married couples. Dump the technobabble, the show isn't about tech, but what tech there is is consistant and make sense. No ubiquitous replicators.
Except for the walled off Romulans and the rarely
seen Klingons, the new crew explores space with NO gigantic powers. Makes first contacts (without the benefit
of the prime directive), fights against minor but strong enemies, encounters pirates and slavers, faces ethical dilemmas, we get to know his crew, this is a time of expansion for the Federation and also Earth.
No one uses the phrase "empire building' but that basically what it is. It's very 15th through 18th century Earth-like.
The Federation doesn't coalesce right away into a more perfect union, there are settling in problems.. This will be less politics and more civilian competition. Trade routes, resources, new colonies -- think of the new Federation as a on-going Journey to Babel.
Humanity wasn't held back for a century by the Vulcans (who are still arrogant assholes), Humans exploded out into the galaxy immediately after Cochrane's first flight. Earth has dozens of colonies, some of which are successful others not.
Throw a little zinger in sometimes, show that 22nd century Humans don't think exactly like us. Two centuries had to have changed us in some ways.
Our heroes lose sometime, sometimes badly.
There is a female Vulcan character, a young minor officer similar to T'Pol, no catsuit. The Vulcans are usually shown in robes of some kind, put Vulcan-girl in something like that. Think about Japanese Samurai/ Jedi Knight form of dress. But still give her freedom of movement. With the right material you'd still be able to see her boobs. Increase the strangeness factor on the Vulcans, Amok Time did a fair job of this. They live 250 years and the average family has 20 odd children, the reason they send them into the desert on the kas-wan is because they want most of them to die.
Junk Star Trek's "modern secular liberalism." Sure a few of our characters will embrace this, but most of the characters are a diverse group in terms of their beliefs. economics,
religion, politics, capitalism,
socialism. The characters are very different from each other.
There would be two secondary characters, one a person of profound faith, the other a adamant atheist. Make them best friends, while it would never be the main point in any episode, there would be a on going C story arc of the two of them arguing, joking, pushing their individual interpretation of the events in various episodes. Think of the friendly arguments between McCoy and Spock. Just something in the background.
Not sure just how "military" the show would be. I think today's audience might accept a more traditional military organizational structure. Or just stay with the para-military Starfleet we've seen. There will be aliens of course, they will be alien through the actors portrayals, not elaborate make-up.
During this particular show Earth/Federation/Starfleet will use filthy money.