There's a difference, Chakotay MAJORED in Anthropology. Picard took some class presumably to fulfill graduation requirements but took a liking to the topic.
We don't know quite that much about Picard's background. He could hold doctorates on a number of subjects; such detail on our hero characters is typically missing, and Chakotay really is a rare exception in the level of exposure he gets.
Also, Picard wears blue all right in the alternate timeline of "Tapestry", but on what merit, we aren't told. Chakotay might have ended up like that in different circumstances, but we have seen versatility and command qualifications in blueshirts often enough, including good old Spock, and supposedly Janeway herself. Even Crusher eventually got a clear-cut Command job and a red shirt.
We have every reason to believe Picard took the shortest route to command, which in the Star Trek world generally seems to be taking that route through the Conn position. We know he has an extensive pilot background that might even rival Riker's and maybe even Tom's.
Does he now? He personally pilots the ship twice, in "Booby Trap" and "In Theory" (once onboard, once using a pilot boat), but attributes this to issues of responsibility rather than skill both times.
Picard's past remains a mystery to us. But what we know of his "direct path" seems to be even more direct than what you are suggesting: he wears Command white right after graduation in "Tapestry" already, rather than the yellow most commonly associated with Helm duties in that era, or the grey that sometimes competes with it. Although we do have one example of a white Helm officer, too, namely Lt. Castillo from "Yesterday's Enterprise
"; perhaps Starfleet practices changed in the early 24th century?
The idea that Janeway was just going on a short mission and didn't bother to get a science officer doesn't fly with me.
A person fluent in sciences, yes. But once that one bites the dust in "Caretaker", a comprehensive science staff might be the thing missing from her crew complement, so no real replacement would be available for the deceased.
Note that several officers at or above the rank of Janeway's XO were listed as dead in "Imperfection". It would be odd to have those in roles where they could take over from Cavit in emergencies, so placing them in positions such as Chief Medical Officer, Chief Engineer and Chief Science Officer would seem prudent. But we have three such officers listed, and after one LtCmdr has been allocated as the CMO we saw, the further LtCmdr and the one Cmdr are still available, suggesting that a CSO indeed was aboard. Or can we invent further "staff" positions that would not compete with Cavit?