It's a good idea.
If the show lifted directly from Blue Bloods and they were all from one family though, this I how i'd imagine it would go...
- The admiral - the family patriarch or matriarch - with his/her inner circle at work. Maybe they are stationed at a Starbase on a far away colony, or they are stationed on Earth.
- The older sibling is a commodore that’s head of Starfleet Security, married to someone at Starfleet medical and has kids with them, and has a revolving door of adjutants
- The middle sibling in the Starfleet JAG Corps raising an only child who want to be a Starfleet officer
- The younger sibling who is a middle decker serving on a starship, and falls for a lower decker
- Another sibling that died in duty as captain and had a long lost child with someone from their days at the Academy
- The retired admiral, the admiral's father or mother, who served in Starfleet many decades ago
Now with the added twists of that admiral is taking down badmirals within Starfleet while having regular therapy sessions with a Dr Melfi-type character, and occasionally butting head with both the Chief of Staff of Starfleet and the UFP President. That might be very interesting to watch.