If it were me I'd go something more along the lines of:
Commanding Officer = Andorian, Thaan (Male)
Considered one of Starfleet's best, having commanded three ships previously, decorated with some of the highest accolades, and responsible for a historic peace agreement with the Romulans. With all that behind him, he was considering retiring from active service to take up a teaching role and help inspire the next generation of officers, but when asked to assume command of the latest Enterprise he couldn't refuse.
Not sure if I want to make the Captain an alien, or at least a full alien. Feel like Andorian would be a bit too odd for new viewers as well. Just something about it strikes me as off. It's a very interesting character concept though.
First Officer = Deltan, Female
Sultry, sensual and sharp as a tack, nothing gets by her. On duty she likes to run a tight ship, ensuring everyone carries out their orders in an efficient manner. Off-duty however she lives by the motto "work hard, play hard". Though she is stunning to behold and has many swooning after her, she is a dedicated Deltan as such her Oath of Celibacy is on file and remains intact.
Deltan didn't feel overly fitting to me, although thinking to it, female first officers are still just as scarce as males I'd say. Think that could definitely be a worthwhile change, and I do like how you described her though.
Chief Science Officer = Nyvari, Female
A species from the Beta/Delta Quadrant border, her species was almost decimated by the Borg centuries ago, but a small enough number managed to evade the Collective and rebelled. They were implanted with a bio-agent that spread through the Borg affecting only other Nyvari, breaking them from the hive mind and allowing them to regain their individuality. Over time the assimilated Nyvari were considered imperfect and ejected from the Collective and the remainders of their species were left alone, but the incident taught them the advantages of combining biology with technology and they began a programme of cybernetically enhancing the remainder of their people, to give them a better chance to survive. This practice continues to this day.
While I still feel very strongly about my Chief Science Officer, I do like your entire idea there, even if it were for a race to be focused on at a time. Very cool idea, and definitely something to think about.
Chief Medical Officer = Tellarite, Female
Gruff, argumentative, surly, and about as warm as a Breen winter, what she lacks in social graces she makes up for with skill. One of the best diagnosticians in Starfleet, she is as much a researcher as she is a healer, with numerous papers and breakthroughs in medical science to her name.
The reason I prefer male is that I see the concept as almost a grandfatherly figure.
Chief Security Officer = Cardassian, Male
His father taught him all he ever needed to know about the finer points of espionage, politics, and sewing. He joined Starfleet to get out from under his fathers prominent shadow on Cardassia and become his own person, though there is still much of the man inside him.
I really like that description quite a bit! Mind if I use it to some capacity?
Ship's Counsellor / Diplomatic Officer = Grazerite, Male
A calm and peaceful man, he has a manner about him that is very disarming. Though he deplores violence, as a serving officer he understands the need for it in defence of others, though he will always try to disarm a situation through words first and foremost.
Bit of an odd species, and almost feels like it strikes a timid/cowardly style. I get what you're going for, but I can't shake that it feels a bit like that.
Chief Conn-Ops Officer = Human/Orion, Female
The youngest member of the senior staff, she had a hard life growing up on a rough colony outside of Federation space where her mother was the only human she knew. Her mother died when she was fifteen and she was left to fend for herself ever since, though she managed to sign onto a freighter as soon as she could and got as far away as she could, ultimately getting to Starfleet, though even there she faced hardships due to the Orion blood in her veins.
But that's just me.
Definitely an interesting final concept, but still trying to picture that. At the very least if an 8th character were possible/8th as a recurring character but that isn't constant, I do like the idea and think it could be used well.
If there is an actual genuine (if somewhat slim) chance of getting this made then I would advise that you then let go of "what feels right" to you and instead focus on what will make the studio the most money, which would mean appealing to as many demographics as possible whilst also trying to make it as original as you can within the established universe, with less focus on "tech that will look cool" and more on characters people actually like and want to watch.
Maybe I'm not explaining the feeling well at all, but I just can't explain it. It's kind of like a part of that special gift, like it says "this is just right!" or "on the right track!". While I understand the importance of focusing on what will make the studio the most money, I also understand the importance of not letting money be the only thing that talks. That's how we've ended up with another prequel series exclusively on a paid service, at least here in the states. I definitely don't want a focus on "tech that will look cool" either, since that feels like a lot of the JJ Abrams style.
I don't know whether any of what I said even makes sense to you, or whether I'm explaining things properly, and maybe I can't properly explain it, at least right now, but I do understand what you're saying and I think I have a general idea of what to do, how to pitch it, etc.