The trouble is, in the current paradigm, the elements they mix in from the past are all things you could reasonably expect everyone to know. They're trying to appeal to a mass audience while still having enough 'member berries for fans, so that's a deliberate strategy.
I don't know about anywhere else, but in my neck of the woods, DS9 is the only Star Trek that never gets re-run. A channel on commercial TV has an afternoon slot for Trek re-runs that cycles between TOS, TNG, and VOY. Once they get through VOY they go back to TOS and repeat. They've done this, day in day out, year in year out, in this same timeslot for the better part of half a decade or more. Never once has DS9 showed up. I wonder why sometimes. Is it because it's a step apart from 'comfortable' Star Trek, of starships and boldly going? Is it because the themes are darker, and the character arcs to intricate for a mid-afternoon syndication slot? Roddenberry always said one reason he eschewed story arc style plotting is because it's harder to sell to the 'after market' of syndication. Whatever the reason, DS9 hasn't got the profile that even VOY does. Sci-fi Channel is the same here, they have a constant rotation of TNG and VOY but won't touch any of the other Trek's at all.
I guess what I'm trying to say -- and I say this as a fan of DS9 -- is that I kind of feel it's too niche to be simply dropped into Picard. The references to it would be just a touch too esoteric for the audience that they're trying to reach. 7 of 9 is ok because practically everyone remembers that Voyager had a 'Borg babe'. Hugh and Data work because TNG still has a profile. But drop Nerys, or Garak, or Bashir into it, and it'd be kind of a 'huh?' for a lot of people, I imagine. At the very least they'd need to be more than cameos, and the episode introducing them would likely need to re-eestablish them for an audience unfamilar with them, and their backstory. 7 and Hugh are easy, they're former Borg drones and everybody knows The Borg. The intricacies and history of Cardassia and Bajor, the rich tapestry of DS9's storytelling, I think it's a tougher nut to crack, without falling into the fan-only hole they want to avoid....