So the issue.. Full systems controlled by a family for over 72h (hell, even over a week) did not become core systems.
In my very recent experience this is due to planet swaps. If you swap a planet with your family member, the timer will be reset. Until all the same people hold the planets in this particular system for over 72h, it will not become a core system.

And even more interesting (or troubling): Once it has become a core system, the core system status will be removed once you swap planets in the system. I am not sure if it has to be a certain amount, but I'm sure it happend due to planet swaps the past few days.

I reported this a while back to Pie, but back then I had no clue what was going on/what was causing this problem.