It was Saru, and it's now Burnham. I wouldn't know how to sum them up - Saru's command style seemed to be very measured as well as empathetic, but at times not decisive enough, like an overly put upon mom who just wants the kids to try and behave. Burnham's biggest problem before S4 seemed to be loner tendencies, which feels like the TV protagonist equivalent of saying your weakness is that sometimes you're
too detail oriented actually, and she seemed to have more or less solved it by S4, but it's possible that I still haven't fully got over my frustration with her character's Mary Sue aspects in past seasons (including S3, where some of the packaging had been improved but she was still the solution to every problem) and that I'm missing the more characteristic aspects of her command. I think possibly what she brings to the table is her conviction - once she sets herself a course, she's going to smile through the pain and make it happen. A bit like the action hero version of Picard from the movies.
There will always be people who will shit on SNW because it has women in it, but it's a dual edge sword because there will always be people defending Disco likes it's the most enlightened and compelling Star Trek ever made solely because it has black people and gay people in it.
Yeah, some people are excessive in their genuflection, and some people are excessive in their blatant reactionary racism etc. Fine people on both sides I'm sure. This isn't a Star Trek thing, though, it's a nerd culture thing, it's exactly the same in Star Wars and superheroes and everything else. I don't know if it's even particularly enlightening to talk about it specifically in the context of Trek. Like you said, it's not really a factor in the writing. (I feel like we're a decade or two out from media that feels the need to explain what gay people are as a plot point, but perhaps that's optimistic.) Disco went hard on the Diversity Space Method, and that's a political choice in itself, but I really do feel very strongly that anyone who's tired of the media shoving black and gay people down their throats, like they're supposed to accept that those people exist or something, really can die mad about it. I'm also pretty sympathetic to people who want to cheerlead for shows like that when those people are the opposition.
In terms of actual political themes in the writing as opposed to casting, I know part of your point is that Disco and Picard are actually
not wildly different from other Trek shows in how they handle political elements, which I don't disagree with. I do think that it's easy to downplay how pointed political commentary was in work from previous decades, simply because those things can't be surprising after the first time and can't be immediate when the present they openly reference isn't this one outside our windows. I do think TOS is unambiguously "preachy" in those cases where the Enterprise encounters a planet with a 20th century social problem and solves it, followed by a speech from Kirk about the moral of the story, and I don't see that as inherently a weakness. (Because I can't dismiss it as a planet of hats with a lesson of the week without acknowledging that the first episode of SNW was deliberately and exactly another one of those episodes, and also a damn fine hour of television.) But I think we might all be reaching at naming the same thing, whatever it is in the writing perspective that allows these things to ring true instead of inducing cringe headaches. It was also pretty preachy and contrived that time explained being gay by having a trans alien decide to be straight for Riker, and there are reasons that's not remembered so fondly.
Whether or not the Federation and future humanity is presented as evolved past our present social bugbears is a very specific issue within that larger space. I think getting Picard S1 just after Discovery had been doing Into Darkness things with a Federation that might be slightly evil all felt like a very big shift in the stories Trek was trying to tell. After Discovery bipped into the future to find an unambiguously good Federation and Picard S2 was just whales, it seems a lot more like those two shows happened to be trying very hard to tell stories that the setting was not built to tell and damaging both the story and the message in the process. And for all the many issues of Disco S4's everyone's-in-therapy plot thread, it did illustrate for me that optimistic stories in better futures don't have to ignore real-world problems, and they can actually model very positive responses to them.
I have had reservations about the "evolved sensibility" concept, but I think they all have to do with TNG. Sometimes TNG really lacked empathy and could be a bit self-righteous, and I think it ran into the issue of being so embedded in a fiction where all the real-world problems have been solved that it couldn't really say much constructive about how to solve them. As long as SNW can stick to a more TOS model as it's been doing, I think it can be a good thing.