But, if Bella's school is assuming she spent more than 15 minutes in close contact with the positive student (were they masked? OP doesn't say.), then Bella's mother and brother are just as likely to have more than 15 minutes in close contact with Bella. So, until it is definitively determined that Bella is not a positive, they should be QT. Because if Bella did get infected, she is going to be able to infect her mother and brother before and if she ever shows any symptoms.
If you wait to find out that Bella is infected, it's too late for Bella's mother and brother, thus too late for their close contacts at their schools.
Most spread is occurring at home with family members and extended family members.
Now if Bella lived alone and stopped at the grocery store on her way home, then the cashier does not need to be QTd.