...They really can't do anything Earth-shattering with Sarek or Mudd. Not sure that Burnham's connection to Sarek and Spock will be anything more than superficial from a canon standpoint...
I don't think this is correct at all. They can't kill Sarek or Mudd, that is true. They can't make all Klingons into our best friends and allies, but they have a lot more room to play than that. The entirety of what we know about Sarek comes from one episode of TOS, one of TNG, and a few lines in the movies. Spock didn't talk of his family at all outside of those (and the final reference in Unification), so we know very little about what Sarek had done or was like. I think DSC has a lot of room here. For example, in Journey To Babel, Spock found it not unreasonable that his father could have murdered someone. That is quite a thing for a Federation ambassador to be capable of, and to be thought so by his own son.
Additionally, the thing about Burhnam's relationship with Sarek that I am most looking forward to is to see what it can tell us about Sarek and Spock. What is Sarek like that he doesn't Speak to his son for ~18 years because he joined Starfleet, but he mentors a human as far as attending the Vulcan Science Academy (VSA) and then encourages her participate in Starfleet? Is his subconsciously trying to make-up for the falling-out with Spock, or is he trying to one-up Spock by raising a full-human through the VSA and into Starfleet. Just what is the deal there? No way DSC chooses Sarek as the father/mentor figure for Burnham without some kind of exploration of it.
As for Mudd, while his portrayal in "I, Mudd" was more cartoonish, in "Mudd's Women" he was darker, harsher, and more villainous. Also, just because Kirk and Spock had to look him up in the computer (which shows only his recorded crimes) doesn't mean that his actions weren't significant - just that Kirk and Spock weren't aware of them.
Yeah, Sarek and Mudd won't be able to shatter known-to-survive-till-TOS planets (ala Vulcan in ST09), but I am sure Sarek could organize a set of interstellar accords (ala Khitomer), or start or end a conflict with an alien race, etc., etc., and who knows what Mudd could get up to.