For the characters in terms of being the lead: The decision-making power lands with the Captain. That's ultimately why Burnham was finally made a Captain.
In terms of characters' experience levels:
Burnham went through four years of the Academy and had seven years of experience on the Shenzhou before DSC even started, and she was a First Officer on the verge of soon getting a Captaincy of her own... until "The Vulcan Hello" and "Battle at the Binary Stars" screwed that up. It took her three seasons to get back to the standing she had at the beginning of the series and finally become a Captain.
Kelvin Kirk went to the Academy for three years, and then leap-frogged straight to Captain. The 2009 Film already had time-skips. If they added a few more, that would've solved everything. But anyway, at least Kirk put in years (plural) at the Academy, took the Kobiyashi Maru, and actually saved Earth and saved the day instead of being saved. Unlike Jack, who had to be saved.
Seven of Nine effectively served four years on Voyager, had up to 20 years of transferrable experience with the Fenris Rangers, had recommendations from Admirals Janeway and Picard, and served as First Officer on the Titan for up to a year (we'll have to ignore that wine bottle at the beginning of PIC S2 saying "2401" and pretend that Present Day for the second season was 2400). So 25 years of experience, five of them on Starfleet ships, and then she's recommended to be promoted to Captain.
Jack Crusher goes through an accelerated program at the Academy in one year, and is made Counselor.
Between these four, Burnham is standing on the most solid ground, Seven works if you allow for transferrable and equivalent experience elsewhere, Kelvin Kirk is extremely shaky, and Jack the most so. Burnham, Seven, and even Kelvin Kirk all had more experience and all were First Officers or Captains. Jack is a Counselor who took crash-courses, had mom and dad pull strings, and isn't an XO or CO.
Jack Crusher, if the series revolves around him, and he's the only one who makes things happen, would be more "Mary Sue" or "Gary Stu" than any other character in Star Trek.