I think this can quite easily be pulled off. Lower Decks is an example of this to a degree, but for an even better Beta Canon example look at the Star Trek Resurgence narrative game. The story takes place from two perspectives Petty Officer and the ship's First Officer. Each of these characters has relationships which will impact the story and for the most part they work separately from one another, but still work together and it makes a lot more sense when the Captain sends the Petty Officers to go on the hull and do dangerous work than sending the chief of any department.
Consider that Deep Space 9's primary cast of characters includes an enlisted person and several non-Starfleet personnel or straight up civilians. Porting that to a TV show would not be that difficult and I think there has even been some success with that in Lower Decks which features lower deckers along senior staff just fine, even interweaving their stories; and Discovery which, particularly in the first season, creates characters by proximity to the story not by bridge positions. Tilly is important because she is Burnham's roommate, not because she's the chief of anything. Despite this Tilly's character is a fan favorite. Unfortunately, I think Discovery fell into the impulse of giving audiences more of what they want and that meant creating stories where a random cadet was a valued member of the team - and team was still mostly senior staff.
undefined> Zipping around the battlefield has virtually no tactical advantage.
This is really the correct answer. If you aren't moving at warp, you might as well not be moving for all the good it will do you. If you are moving at warp you're moving in a straight line so quickly that unless the other ship is following you at warp, it doesn't really matter.