A sensible rule would be to require liability insurance, as with driver’s licensing in most States.
As with other non-licensed activities, any damage you can do in your daily, non-professional life is normally covered by your civil liability insurance. Not sure about the US, but in many other countries, it's often covered in your home insurance. So, until drones are actually licensed - which I don't think this certificate will be, at least for now and strictly speaking - any damage should be covered by civil liabilty insurances. Just like if you cause damage with a frying pan. Or any toy.
We don't have any such thing as a generic "civil liability insurance" here in the US.
If you own a house, you can take out a homeowner's insurance policy. (If you have a mortgage, the lienholder requires a homeowner's insurance policy.) The insurance covers obvious things like replacement in case of fire or other calamity (flood insurance is separate!) as well as theft and repair of damages (like if a water line bursts and destroys your floor and cabinets). Theft coveraged is extended to losses outside of the home, for instance if your kid's laptop is stolen at school.
The policies usually include a "personal liability" component, which covers you if, for example, someone slips and falls on your property and is injured. This policy does
not cover you if you fly a drone in a public space and you crash it into someone.
It's why automobile insurance has a liability component. If you are driving and cause an accident, the insurance covers you for the liability. The liability part of your homeowner's insurance won't cover this.
Now remember that not everyone owns a house! If you don't own a house, you're not required to have homeowner's insurance, so therefore there's no possibility of a "civil liability" insurance. Renter's insurance covers your personal property stored in a rented home, not the structure itself. Insuring the structure is the landlord's responsibility.
The whole idea of liability insurance is to ensure that a plaintiff won't take your house and other property if you are sued and lose.