The writer of that episode of DS9, Ron Moore, has said it was a screw up.According to one episode of DS9 the Eugenics Wars happened around the year 2173. And in early ENT Archer grew up around San Francisco - until later episodes had him growing up in New York State.
MOORE: "This is my personal screw-up. When I was writing that speech, I was thinking about Khan and somehow his dialog from 'Wrath' started floating through my brain: On Earth... 200 years ago... I was a Prince... The number 200 just stuck in my head and I put it in the script without making the necessary adjustment for the fact that Wrath took place almost a hundred years prior to Dr. Bashir. I wrote it, I get the blame."
Fine. Then the appearance of the USS New Jersey in Picard season 3 means all of this mess doesn't exist, and by that point in the 25th century it was corrected to the TOS timeline, if we go with the idea that everything is mutable and the continuity can be whatever we want to believe it to be.Same way Wrath of Khan fits with every other Trek series that changed the date of the Eugenics Wars. If you need to make up some kinda head canon that it was changed by the temporal cold war or something, go right ahead.