OK, it's important to step back from thinking of the Trek universe as existing separate from those who guided it and just accept that there was a sort of push and pull through all Trek's incarnations based on the powers-that-be.
Gene's vision of Trek de-emphasized rank. That's why TOS uniforms don't even feature braids for ensigns, which maybe went a little too far. When you get to TNG, one ranking pip indicates ensign.
Nevertheless, even in TOS there was some sense of pecking order and pomp and circumstance, hence the satin dress uniforms with the triangular campaign pins, but I've read that Gene didn't even like that.
Consider that the battle in Khan (aside from the tall ships homage) hearkens back to Balance of Terror, which in turn is sort of a remake of Run Silent Run Deep. And therefore it is VERY militaristic. So you can't say Trek had no militaristic overtones. It's just that this was played up or down at different points in TOS depending on who wrote the script or exerted the most influence.
When you get to TMP, Gene was allowed to establish the main tone of the Trek universe, and he struck a tone that didn't really return again until TNG. While Picard was very stiff at first, and one for discipline, the Enterprise as a whole was supposed to transcend the military and be a city in space with married families and kids. The Enterprise-D bridge has a very lounge-like vibe to it. With the woodgrain panel in the back and the earthen tones, you'd almost expect to see a fireplace in the corner and kids playing with blocks in front of the CONN.
I think thoughts are pretty well divided on whether Starfleet life should be by-the-book or the more politically correct "let's sit this down and talk this through, shall we?" that Gene preferred.
The reason the Nick Meyer era is preferred is that, left to his utopian ideals, Gene's world is rather dull. So yes, the red-coats (I really see them like revolutionary war red-coats, although some people think they look like mounties) are ornate, but this was basically a reaction against the short-sleeve T-shirt aesthetic that was struck in TMP, and maybe in retrospect, an overreaction.
On a practical level, it also bears noting that in the movie era, the actors were starting to lose their physiques. Doohan was the first, who seemed to put on a lot of weight between TMP and Khan, followed by Nichelle Nichols and Shatner. The woolen red uniforms helped to disguise this better. There was no way anything resembling the TMP uniforms could have been carried all the way through to Trek VI without the entire Trek cast being ordered into the gym.
Of course, none of this actually reconciles one coherent universe where everything makes sense, but that's what you get when creative forces shift from one power-center to another over the span of decades, and now you have JJ Trek with its own quirks to argue about.