I think Picard would accept anointment as Emissary, probably reluctantly, if offered it.Regarding the question in general, I would say DS9 is the easiest to answer: Neither Picard nor Janeway would do half as well in Sisko's position as he did. The first major obstacle is, of course, the Emissary. If Picard and Janeway aren't the Emissary, then a huge portion of DS9 is undone, including much of the cultural exchange that brought Bajor closer to the Federation (DS9's original mission) and one of the most pivotal moments of the Dominion War (DS9's later mission). Even if we rewrote their characters to say that they actually were the Emissary, neither one of them would ever really accept that position and thus would still never achieve the same level of relations with the Bajorans or the Prophets.
I also doubt if Picard would truly understand the importance of the civilian life on the station or the continuity of that civilian life even between different regimes. Under Picard, Quark would've been gone from the first episode and most other civilians with him. DS9 would've been almost exclusively a Starfleet/Bajoran Militia station guarding the wormhole but not really attracting any sort of cultural exchange. I'm not sure where Janeway would fall on this question, but I'm guessing maybe closer to Sisko than Picard.
Picard accepted the position of Klingon Arbiter of Succession, although he was initially reluctant. Picard was persuaded to accept because of some "doing it for the greater good of Federation Klingon relations" type of argument from K'mpec. Later on, Picard admitted that being Arbiter was a conflict of interest, but he ultimately didn't refuse to take on the role.
As with the Arbiter situation, Picard could be persuaded to take on the position of Emissary with the same type of "fulfilling destiny and for the greater good of Federation Bajoran relations" argument.
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But you're probably right that Picard would have run Quark, Garak, and many civilians out of DS9. Picard wouldn't put up with such shady characters. He would rent out Quark's bar to someone like Guinan -- a boring, no fun, straight arrow bar tender. Gone would be the Dabo games and other such activities. DS9 would be a sleepy backwater outpost under Picard.
Also, I could see Picard having a very rocky relationship with Kira and Odo. Kira and Odo are strong willed personalities. Both Kira and Odo speak their minds and they both defend their turf. There would be a serious clash of personalities.
Picard is too rigid and by the book, as you described. Picard wouldn't be able to so easily get rid of those two or bend them to his will because they are not in Starfleet. Picard would not be a good fit for DS9.