Thoughts from the perspective of a "progressive liberal SJW".
I'm not a hardcore Star Trek fan. I'm only a Star Trek: The Next Generation fan. I haven't watched the other Star Trek series so keep that in mind.
I'm also not the show's primary target audience. I don't watch network TV shows, my standards for what good TV shows should be are painted by shows like Game of Thrones, Westworld, House of Cards and Black Mirror. So, mileage may vary for people who are fans of network TV shows and who have (in my opinion) lower standards. This may come off as elitist, but recognize that I fully appreciate that different people have different tastes. Furthermore, it is my belief that elitism has benefits if it can push shows to improve. If shows like Game of Thrones, Westworld and House of Cards and Black Mirror can push network TV shows to be better and have better writing, it's a benefit to everyone.
Now, on the subject of writing. Star Trek: Discovery's writing is certainly not the worst I've seen. The writing and dialogue is CERTAINLY not as bad as the writing of Lost (puke). There are some odd dichotomies present, however:
For example, Michael is meant to be pseudo-Vulcan, given her Vulcan upbringing. (I have to reiterate at this point that I'm only very familiar with The Next Generation, which did not feature a Vulcan crew member so my knowledge is limited, the closest anchor point for me is Data.) As a person with a Vulcan upbringing, Michael should display far more Vulcan-like personality characteristics than she exhibits. Now, I understand she is human and therefore does not have the emotional muteness of a Vulcan something which is presented by the dialogue clearly. But, her displaying human emotional volatility and mannerisms all the time renders her Vulcan upbringing close to irrelevant aside from the cheap surface characterizations of knowing Vulcan nerve pinching techniques and being highly educated.
The scene between the former captain and Michael at the elevator after they first meet is a stroke of genius, I will grant this. With just a couple lines it develops their relationship by leaps and bounds. This is something I appreciate. I just hope Michael maintained more of her Vulcan-ness with time.
A lot of times Michael comes off as cold and snarky. Doubtless this is the way the writer's decided to show her Vulcan side. But, from my perspective Vulcans were not meant to be snarky, instead their coldness and propensity to be emotionally distant comes from a genuine lack of understanding or even a overly-logical lack of appreciation for human social customs.
In The Next Generation, the crew members and characters never (or very infrequently) made snarky remarks or remarks showing overt malintent. Humans in general are much kinder to eachother and there is a general sense of agreeableness and very little tension between between characters. I always assumed that this is because in the future people just learned to behave kindly towards one another. The characters in The Next Generation are almost inhumane by our today's standards. This inhumanely agreeable property of humanity is clearly not on display in Discovery. Infact, quite the opposite is on display. You can tell this from the Europa captain's behavior towards Michael. Another example: in episode three, the security officer says "I see we are unloading all kinds of garbage today" referring to the prisoners being transported. This line was very shocking to me. Certainly you wouldn't see this kind of dialogue in The Next Generation.
All that being said, I understand that The Next Generation took place far in the future from Discovery's timeline so maybe humans learned to behave nicely towards one another some time in the intermediate.
Regarding the Klingon's: I really liked the redesign. The Klingon social hierarchy was displayed very effectively with the emphasis on familial houses and dedication to house honor. I also thought the Klingon motivations were very well fleshed out. A warring race deprived of the opportunity to war for many years and tensions bubbling under the surface until they finally find an excuse to war and conquer again.
A lot of people have found analogues between the Klingons and white supremacists, but I think you can equally find similarities to ISIS and al-Qaeda who similarly want to preserve their nefarious traditions and culture. It's sad to me that people will find an excuse to come to the defense of white supremacists and shout liberal propaganda at the Klingons instead of analyzing the deficits in their logic. Maybe if a culture is built on and devoted to war as the Klingon culture is, or maybe if a culture is devoted to marginalizing women and maintaining a biological social order as the Muslims culture is, or maybe if a culture is devoted to looking down at and taking advantage of "lower" races as the Confederate culture is, then maybe those cultures are not worthy of being maintained.