I like the keeping historical records part. But an important part of being in a family is face to face problem solving. Then again it might allow family members with little voices to be heard instead of being stepped over.
I was also thinking about onboarding, lol. New family members would be encouraged to look over old tickets to help with the legacy problems that will likely never be fixed.
Jira might fit well with all its "issues" and such, (sorry being cheeky :)
Another, more light weight platform would probably make the family like it better.
I actually thought about Trello for family issues. With two kids it feels like managing a startup. Did you mean something in that nature?