After looking through this thread, I note that nobody has noticed the Denobulan Mammoth in the room.
Consider the time frame: Mid to Late Pike era. A year after The Cage.
We know, from one of Spock's lines in The Menagerie, that well over a decade passed between The Cage and The Menagerie. We can also surmise that a fair amount of time passed between Where No Man Has Gone Before and The Corbomite Maneuver, from the fact that the Galactic Barrier made a real mess of the Enterprise, and from the uniform changes (e.g., services going from tan to red, black collars, the introduction of the "skant" for female crew).
In The Cage, the standard Starfleet issue sidearm is the "hand laser." What at least one fan publication elaborated on as the "40mm hand laser." In Where No Man . . . , the hand phasers appear to be rebuilt hand lasers.
So why, in the cover painting, is Burnham, from a ship that was already old when Pike was commanding the Enterprise, holding a recognizable TOS-era type-II hand phaser, a model that probably didn't exist in the mid-to-late Pike era?