I re-read the Character Breakdowns and, if I were to hazard a guess, I'd say Picard's mission isn't above-board. Given who he's associating with, I'd say this is a mission he's taken on his own accord. If he were still in Starfleet, it's probably something they wouldn't approve of. I think, like I might've said before, Picard hired contractors to help to get him where he needed to be.
Connie and Lawrence seem like people Picard brought on, no questions asked, because they could get the job done.
Alana, being a former intelligence officer, probably knows stuff that can help Picard get around.
K'Bar is probably a Romulan from Spock's underground who is now loyal to Picard who might be picking up Spock's mantle.
I have no idea about Straton but Dr. Smith is probably brought along with him. Or maybe Dr. Smith comes with the ship since holographic people have probably become standard on all ships since the VOY Era, and not just on Starfleet ships.
Indrina, I'm guessing, is an aspiring young archaeologist who's really focused on archaeology to such an extent that she's socially inept because she's not interested in anything else. Maybe Picard took her under his wing and remembers when he was thinking about considering a career besides Starfleet. And now here he is doing exactly that: something besides Starfleet.
Picard. I don't know how much DS9 will be addressed, if at all, but I've been of the belief for 20 years that the Dominion War changed the Federation. We see so little of the Post-DS9 AQ and nothing of it that we do see really challenges that idea. They don't need to refer to the Dominion War per se, but they can say that Starfleet has changed for the worse. Maybe the Federation, as the only standing major power, has also changed for the worse. He and Starfleet no longer see eye-to-eye even though he thinks what he's doing is in the best interest of the Federation. It would make him more willing to go with a "rag-tag group" if he had very little support from anyone else. I'm guessing he and Crusher are no longer together. Someone else suggested she might've died. Which definitely would've put him in a funk if that's the "different place" we find him in at the beginning of the series.