What about T'Kuvma's followers? Are we really to assume L'Rell and Voq were the last? I would think some Klingons would hack into everything they knew about Burnham and then get revenge. If not as of Season 2 Discovery, someone would be digging it up during the Organian War.Yes, Michael Burnham was somewhat infamous after the war, but I never got the impression that her relationship to the Vulcan Ambassador Sarek was all that well known outside of the corridors of power on Vulcan and a few close Starfleet personnel, or you would expect for there to be some blowback for him or mention of it after she got imprisoned. And Vulcans by nature keep things close to the chest, so I wouldn't expect them to talk about it much. Sarek would probably want to keep the relationship low key outside necessary exchanges like the Vulcan Science Academy, especially after the Logic Extremists targeted Michael and the family.
So in terms of people questioning Spock, since they didn't even know his father was the Vulcan ambassador or that he had a brother, there probably shouldn't be a reason for her to ever be brought up in his presence anyway, since regardless of whether people know of the infamous Michael Burnham, very few would associate her with Spock. Discovery is gone, there's no reason most of the Enterprise crew besides the bridge officers has to know they're brother and sister.
Klingons do a lot of hacking. That's how they found out about Genesis.
Remember Burnham didn't just kill T'Kuvma, she killed Kol, of House Kor, and his ship in a battle where she dishonorably beamed out. Klingons don't forget things like that. That would very much be of interest to Kor in TOS, and Spock would be a target.