When it comes to Discovery’s take on diversity, we’ve never seen a black woman as prominent (including Uhura even) on a Trek series/movie as Burnham, so that does make the show stand out. (It would be great if they did something though to root her more in a culturally black experience to demonstrate the difference, though the visual makes a statement on its own. A sci-fi series where the main character is a black woman. The character the audience is supposed to identify with, follow on their journey, watch their lives and loves, victories, and defeats, etc.; Off the top of my head I can’t think of another sci-fi series that has been led by a black woman). After Season 1 alone, Burnham is the most developed black female character in Trek’s history.
I suppose it is true that Trek hasn't had a black female lead before. But Trek has had a black lead and a female lead separately.
As for rooting the show in a culturally black experience - DS9 came about as close to this as possible with Far Beyond The Stars in particular, but this had to be done through a 20th century setting. The thing is in the Trekverse, human racism has vanished. So much of the black experience as it is understood in the west - particularly the U.S. is rooted in a response to racism. Take racism away, and everyone would just start to culturally blend together. Actually, genetically blend together too. One of the big missteps of Trek was that in a world where even inter-species children are common, and interracial marriage is completely normal, there should be a lot of multiracial people and relatively few humans left who look any one thing in particular. Of course I recognize that Trek is not supposed to represent what the future will actually be, but something that contemporary people can relate to. Still, it would be irresponsible to depict a culturally authentic black experience shorn of racism, and (human) racism canonically doesn't exist in the Trekverse - so it's better to leave it mostly unspoken.
Also, while as you’ve mentioned, there have been some exploration, perhaps on the edges, when it comes to LGBT issues/characters, there has never been as prominent a homosexual character as Stamets and we have yet to have seen as much exploration of an LGBT relationship as Discovery has depicted. Perhaps this is just me, but when it comes to Georgiou, Cornwell, and Tilly, as well as Burnham, it feels like Discovery is making female characters more integral to the show in ways that feel different-to me at least-than some of the previous Treks. I also think Discovery has taken some steps to make their crew feel even more inclusive, in terms of background players, than I’ve noticed in previous Treks.
I do give Discovery credit in that women are finally being written better. Even during the Berman era, Kira was the only female character which wasn't written with her femininity in some way central to her character. Nana Visitor even mentioned when she auditioned for the part she was astonished that the part wasn't written as "the woman" but just an amazing character that happened to be a woman. All of the female characters on Discovery feel like this. They aren't meant to be "the babe" or "nurturing" or anything else. They're just themselves. Now if only they could flesh them out a bit.
I see Act 2 as when Burnham starts to grasp the lessons she’s learned since the Battle of the Binary Stars. This is where they demonstrate her growth. I can see why she attached to Mirror Georgiou. I mean she’s got to be feeling very guilty about Prime Georgiou and she’s lacking a mother figure, mentor, and friend. (One thing that I did catch while watching the show was that Amanda and Burnham seemed to have a good relationship, so I don’t know why Georgiou would need to be a substitute mother like the show set up. If Soval had been the foster parent instead of Sarek I think this would’ve made more sense, IMO. I go more into why I think Soval and/or T’Pol would work better instead of Sarek later on). I think Burnham throughout has struggled to try to reconcile her human emotions with her Vulcan upbringing, so she doesn’t have ‘normal’ reactions at times to things. It reminds me a little of when Data in Generations, though not played for broad laughs like that. Burnham has struggled, is struggling with her divided halves, but her brilliance and ability to get the job done make her valuable to Starfleet so they have been willing to work with her.
Burnham feels like she was sort of focus-grouped in a lab or something. The writers realized that the breakout characters in each show - Spock, Data, Worf, The Doctor, Seven, etc - were torn between two worlds in some sense - they were outsiders to humanity looking in at us. They tried to merge this with Tom Paris's backstory. In theory it could have worked well, but I feel like the singular focus on her - along with direction not to emote heavily due to Vulcan upbringing - alienated a lot of folks.