Florida's new "anti-rioting" bill does this and I think this is one of the better parts of the bill by far. Back in November the original bill legalized shooting looters
https://www.complex.com/life/2020/11/florida-governor-drafts-bill-allow-people-to-shoot-looters-riotersThe draft explicitly mentions "rioters" and "looters," allowing citizens to use force against people who might be looting within 500 feet of a "violent or disorderly assembly." This could make protests especially dangerous in the state, where outside agitators have been known to incite violence and theft.
This part as far as I can tell was left out. So my opinion is if the government doesn't allow one to protect their own property then they should assume defense of it and on paper this sounds like a decent idea.
On the other hand it depends on the effort made. I think what happened at CHAZ should clearly mean the local government should be liable although if the issue was that the police tried but couldn't stop the looting then maybe they shouldn't be liable.
Obviously in the end any final payments should come from rioters/looters.