Of course I would never trade TNG for "Phase II", especially not Picard for Kirk.
(In fact, if I had my I'd remove some more of the left over Phase II elements, like Riker and Troi being ex-lovers and such, and Riker being the guy who waits in the wings to replace the Captain)
But what could have been interesting would have been setting TNG right after The Undiscovered Country, maybe five or, at most, ten years after it (I know, impossible, because the film came long after Season 1 of TNG, but I'm talking about the time period).
I would keep the look of TNG (the new uniforms would be explained as Starfleet trying to look less militaristic after peace with the Klingons) and I would keep most, if not all of the characters with maybe a little tweaking (the TOS characters could have occassional cameos as statesmen or, in the case of Uhura, for example, intelligence agents/contacts)
But instead of trying so hard to distinguish itself from TOS in terms of world building, I'd try more to build upon what was established in TOS.
For example, I'd might make Dr.Crusher a Vulcan or a Half-Vulcan. Troi would be a telepathic human like Miranda Jones from "Is there in truth no beauty?" (In fact, maybe Diana Mulaur could appear as Miranda Jones and be something like Troi's mentor, or instead in a more serious version of Lwaxana as Troi's mother, just say the telepathic humans also tend to live longer) I'd have a recurring Andorian character and have the Orions work with the Ferengi. There'd be a Gorn episode in Season 1.
A big thing would be the change in the Klingons, rather than the "wargh we am basically rowdy space bikers, we eat living worms hurhurhur" guys they devolved into as 90s Trek went on, I'd keep them more in character to how they were portrayed in TOS (though I'd keep the makeup from TNG/the movies)
And I'd portray the Klingon Empire being in the process of completely falling apart ever since the peace treaty. many resource rich worlds populated by subjugated species would have declared their independence under the protection of the Federation, several Klingon worlds would have seceded from the Empire (either going rogue or joining the Federation) with the Empire reduced to a rump-state of only the oldest core worlds (which are resource poor, as stated in Day of the Dove) and little more than a protectorate under Federation control.
This would allow the exploration of many interesting elements (in my opinions), including themes around refugees and immigration (a wave of Klingons leaving to settle in Federation space) plus the threat of the Orion-Ferengi alliance exploiting the chaos and lawlessness and the Romulans moving in to claim new territory (there could be an episode where the female Commander appears again, or several). And of course there could always be a remnant of the old Klingon Empire somewhere at the far end of space that tries to re-establish their old power and wage conflict with the Federation again.
And yeah, as
@Kor pointed out, what to do with Worf in this scenario is an interesting question.
I think my favourite solution would be to have him one of the Klingon refugees/immigrants into Federation Space after the peace treaty, either as a young, disillusioned warrior, or as a youth with his parents. That way a lot of that thematic could be explored through his character directly. He'd probably be someone who'd hold onto as many traditions as possible, while K'helyr, on the other hand, would be a Klingon immigrant who has completely immersed herself in Federation culture.
