Star Trek: Discovery's Michael Burnham (Sonequa Martin-Green) took three seasons to become a Captain, but for different reasons than why it also took three seasons for Benjamin Sisko (Avery Brooks) to become a Captain on Star Trek: Deep Space Nine. Burnham and Sisko are the two African-American leads of their live-action Star Trek series, although the animated Star Trek: Lower Decks also has an African-American main protagonist, Ensign Beckett Mariner (Tawny Newsome), who has no interest in becoming a Captain.
Benjamin Sisko was a breakthrough character as the lead of Star Trek: Deep Space Nine, and yet, he was introduced as a Commander. Instead of following in the footsteps of William Shatner's Captain James T. Kirk and Patrick Stewart's Captain Jean-Luc Picard, DS9's creators and the studio, Paramount, opted to give the first black lead of a Star Trek series the lesser rank. Their logic was that a starship is led by a Captain, but DS9, which was set on a space station, had to have a Commander in charge. (This despite Star Trek: The Original Series and Star Trek: The Next Generation establishing that Federation Starbases were run by Commodores and Admirals.) DS9's executive producer and eventual showrunner, Ira Steven Behr, totally disagreed with making Sisko a mere Commander; although it took three seasons, he and Avery Brooks won their battle to promote Sisko to Captain, the rank he deserved.
Michael Burnham's circumstance in Star Trek: Discovery was different. The series' original creator, Bryan Fuller, initially intended for Star Trek: Discovery to be an anthology series with each season being set in a different era of Star Trek. Another innovation by Fuller was that Star Trek: Discovery's main character wouldn't be the Captain, which would further differentiate his show from the tried-and-true Star Trek model and give a fresh and different point of view to the show's adventures. Even after Fuller left and Alex Kurtzman took over production of the series overseeing different showrunners, the POV of the series remained Michael Burnham's, who was introduced as a Commander (like Sisko), and who even controversially began the show by becoming Starfleet's first mutineer. As if further mocking the idea of the Captain as the center of their series, Star Trek: Discovery began an odd game of musical Captain's chair, with numerous characters becoming Captain of Discovery throughout its three seasons so far.
It's certainly an unusual coincidence that both Sisko and Burnham weren't Captains of their Star Trek shows from the get-go but both situations ended up working out for their characters and their respective series. Sisko was a different, more somber character in the first two seasons of DS9 but fans got to watch him gain confidence and greater authority as DS9's leader and as the Emissary of the Prophets. Coinciding with Avery Brooks being allowed to shave his head and grow a goatee, the new-look Captain Sisko marked the creative galvanization of DS9, as the show got more serialized and more daring in its character arcs and storytelling. While not fully appreciated when it was on the air, two decades later, DS9 is revered by Star Trek fans new and old, and properly regarded as an innovator of the modern serialized Peak TV format.
Certainly, without Deep Space Nine, there would be no Star Trek: Discovery or Michael Burnham, which continues DS9's serialized, character-driven format as opposed to TNG's classic, episodic style. Further, Burnham's rocky road to becoming Captain really began in Star Trek: Discovery season 2, and showrunner Michelle Paradise told Wil Wheaton on The Ready Room that the decision to pull the trigger on Michael's promotion was made at the start of writing season 3. This made Burnham's voyage to the Captain's chair the underlying point of her season-long arc, with Michael initially unsure of her future in Starfleet before a brief, unsuccessful stint as Captain Saru's Number One. By the time Burnham achieves the pinnacle of her career, her Captaincy is a deserved achievement for saving the Federation, and Michael further earned the respect of her crew and superiors.
Captain Burnham now joins a legendary list of Star Trek Captains who headline their own series, which includes Captain Sisko, who blazed the trail. Further, Burnham also becomes the second female Captain starring in her own series, following Star Trek: Voyager's Kathryn Janeway (Kate Mulgrew). Although Burnham and Sisko had a longer road than their peers to get their Captaincy, it ultimately made joining them on their journeys towards their destinies on Star Trek: Deep Space Nine and Star Trek: Discovery all the more rewarding.