Since this is a "headcanon" thread: Yes. For me DIS also takes place in an alternate continuity. Same as the Kelvin movies.
For me, purely due to the in-universe worldbuilding: I do not buy that no one tried to dig up the mushroom jump technology, to save the Voyager, or during the darkest days of the Dominion war. The Klingon war, Mudd, Spock's sister, the general aesthetics of everything also don't match up with TOS.
SNW then - which I dearly love - is a weird one. For me it's also in a separate continuity, where "a" version of Discovery has happened. And in TOS "a" version of SNW happened. But it's clearly too different to have happened in the DISCOverse. However - for me Nichelle Nichols is clearly also not the same Uhura as Celia Rose Gooding. And if Vulcan divorce is as easily possible, "Amok Time" makes no sense anymore, and I'm not sold on Paul Wesley or the whole "Ilyrian" thing either.
However I have transferred A LOT of Pike characterization & adventures to Jeffrey Hunter - so basically, for me, SNW takes place in both the DISCO & the Prime universe - EXCEPT the parts where it contradicts either!
So, all in all, SNW is - for me - actually the show with the shakiest "canon" status. It's also one of my favourites. So proof, that this doesn't really matter that much.