This may be too much of change to meet the OP’s requirements, but…
1.) 1st officer and 2nd officer are more than just people who sit at spots on the bridge when the captain’s on deck. They actually take watches and are in command on a daily basis. In fact, there would be a 3rd officer (mate), in all likelihood, too. On a giant city-in-space like the Enterprise, I would see the captain as having almost no bridge time, while 1st, 2nd, and 3rd are the ones making basic decisions, only calling the captain in from his office for emergencies. (Actually controlling things like away teams from bridge.) Otherwise, he’s (as stated upthread) mayor of Enterprise-ville, enforcing the rules, ensuring training is done, adjudicating squabbles, etc. Counselor is his primary assistant in such matters. Away teams may be led by 1st - 3rd officers. Diplomatic teams will always be led by the Captain or, in rare-cases, 1st officer.
2.) Agreed that Tactical Officers, Chiefs of Security, and Marine Commanders are different roles. Yar stays on as a T.O.; Warf is a marine officer. Chief of Security would be more like Chief of the Boat; highest NCO. Which brings me to:
3.) Non-Commissioned Officers. With an infrastructure as large as is necessary on the Enterprise (let alone the rest of the fleet), you’re going to need more variations than O-1 through O-6. If you want to make things more futuristic, make warrant officers the junior officers, and have them do the actual manning. (helm, communications, weapons, engineering, shuttle-craft piloting, etc.) NCO’s from private to sergeant [yeah, I know I’m freely mixing ranks from different services] can be the, well, grunts. IOW, have a real command and rank structure, not just the one where everyone’s an officer for no good reason. Maybe make it so that everyone starts as a buck private for 2 years, and only then is eligible for Academy.
4.) No families on board. Period. Who in their right mind would want to bring their kid on a military vessel? If you want to show that the captain doesn’t like kids, there are plenty of other ways to do so. So, to show human involvement:
5.) More with the hobbies and life not in uniform. No-one is going to want to be cooped up for a five year mission on a ship (even with a holodek). Somebody’s gotta have leave at some point. Show family then. And have it be a mixed bag of real people. (Tactical officer goes home to a partner, has wild-and-crazy I missed you sex for a week straight, then they start getting into fights until leave is over.)
6.) No sports? In the holodek? Pull the other one, it’s got bells on it. The monthly game between divisions would be a huge event. Engineering versus Marines. Science vs. Tactical. Switch the games around. Invent new games. Bring in games from other cultures.
7.) Also agreed on trying to make aliens more alien. Yeah any human drama is ultimately going to be about humans, but don’t make it so blatantly obvious.
OK, so I guess I’ve completely ripped up the show. To put it back together again, you would have to follow a group of young-ish officers as they make their way through the ranks over the span of the seasons. Riker is a fresh faced 3rd officer who befriends a bunch of other junior officers (Warf [marine], Geordi [eng.], Yar [tactics], Data [science], etc.) Picard can stay on as Captain, but has to actually take leave once in a while. (Likewise Dr. Crusher.) Wesley is a warrant officer fresh out of Academy. Other senior officers come and go. The main crew actually learn things throughout the show, becoming more of the supermen and women that TNG wanted them to be. Barkeley could be someone Geordi gets promoted over.
Each year is a season. By the middle of the fourth season, all of the above have been promoted to their “classical” positions. At the end of fifth season, Riker commands a new ship and takes most of his cadre with him. He’s an ambitious sonuvaphaser. (Possibly, the crew move onto another ship after the Enterprise has been destroyed.) There is no way that he would turn down command of a ship. Make Picard the Admiral for the Exploratory Fleet, with Data as his 1st officer (basically running things, but always in Picard’s shadow) on the Command Ship. Riker has Yar as his 1st, Geordi as his C Eng O, Warf as Marine Commander, but on a smaller and more military ship. (A Protector in the Exploratory Fleet.)