I would just say this: I find the decision to set DIS ten years before TOS to be puzzling and kind of arbitrary. There's no compelling storytelling reason why it had to be set in the 2250s -- the core of the story they were telling in S1 (Klingons taken over by extreme nationalism, Federation values of multiculturalism winning out, Burnham acts out of bigotry but learns from her mistake) could have set in, say, the 2390s. The details would need to be different (e.g., instead of Burnham's parents killed in a Klingon raid in the 2230s, you could say they were killed during the UFP-Klingon War from DS9's fourth and fifth seasons), but the broad story would be the same. It goes without saying that the story of S2 could have been done with only minor tweaks as a 2390s-era story.
The characters don't really gain anything from the TOS connection -- Sarek and Spock could have been any Vulcans in terms of how they relate to Burnham. Sarek is just a generic "stern Vulcan with a secret heart of gold" type. Spock's eternal "Human/Vulcan" conflict barely registers until the end of S2, when suddenly he's developed a catharsis to his internal conflict that seems... about 15 years too early given what we saw of his internal conflicts in TOS and TMP. Amanda is only barely a presence until a few key scenes in S2. And having Burnham be this previously-unseen Spock sibling retroactively creates this weird subtext to Spock/Sarek scenes from the prior shows. Sarek, Spock, and Amanda could have been different Vulcan characters and the dynamic would mostly have been the same.
For the most part, Pike could have been any generic Charming White Guy Leader Dude. Really, the only time a DIS episode benefited from and used its TOS connection in a meaningful way was "If Memory Serves" and Pike's vision of his fate from later in S2.
And, yes, it would have been a side-benefit if setting the show post-VOY instead of pre-TOS meant that it didn't have minor continuity hiccups like the spore drive being around 120 years before Voyager got trapped on the other side of the galaxy, or the ship designs generally all looking like they evolved out of the John Eaves-designed ships of Star Trek: First Contact instead of fitting into the TOS visual aesthetic, but that's a side benefit rather than a direct reason to do it differently.