I'm not talking about the entire concept, obviously. I'm talking about the parts left over from 1960s or earlier pulp sensibilities, the parts that haven't aged as well as the rest. Stuff like a crew dominated by white men with only the odd token otherwise. In an "alternate timeline" TOS like Kelvin, one that's supposedly branched off of the same origin, the characters need to be the same people with different lives, so not too much can be changed. But in a wholesale reboot, you could have a female Kirk, a black McCoy, a cyborg Scotty, whatever you wanted. You could erase the chauvinism of the original completely rather than having to tiptoe around it while pretending it's still in continuity. There are also outdated ideas like the relatively primitive medicine in TOS, a grasp of neurology and genetics (and forensics, in "The Conscience of the King") that's less advanced than what we have today, let alone 300 years from now. There's the lack of nanotechnology and genetic medicine and other advances that will be commonplace long before the 2260s.
And of course there are the "near-future" historical events that have already not happened. We're 20 years past when the Eugenics Wars were supposed to occur. We're supposed to abandon interplanetary sleeper ships for faster space drives next year, in 2018. We're not far out from the Bell Riots or the Ares IV flight. The more time that passes, the more explicit Trek history is going to become outdated. What happens to Trek when it's actually 2063 in the real world? If it hasn't been rebooted by then, it's going to seem like a completely antiquated view of the future that never came to pass. If it's to remain relevant, sooner or later it'll have to start over with a new continuity that builds forward from the present rather than the 1960s.
The whole value of adaptations and reboots is that you can be selective in what you keep and discard. You can keep the core elements that are important -- the characters, their relationships, their values and goals, their overall situations -- but discard the side elements that age poorly and end up working against the intended message of the story. It's the same kind of updating that's been done to stories throughout history, keeping their essential ideas alive by adapting the elements around them to fit a new era and a new audience.
Yes, but more viewers today probably associate the pre-TOS era with the blue jumpsuits from ENT and the Kelvin uniforms from ST '09 (which had three colors but used blue for command) than with the drab turtlenecks from the pilots. The older generation of dedicated TOS fans is nostalgic for those drab turtlenecks, but to the mass of people in the general public, ENT and the new movies are probably more familiar than the original pilots. And the DSC uniforms seem like they could be plausible descendants of those uniform styles.
Anyway, most viewers don't care that much about continuity details. They just want to be entertained. They want the stories to be engaging and they want the show to look good. As long as the uniforms look good, most audiences won't care much about how they line up with previous shows' uniforms.