I think we would see far fewer actions at all for fear of being sued.
As it currently stands the police already do almost nothing. Any kind of push back or critique of the police leads to inaction by the union. Meaning, police twiddle their thumbs and take your tax money because they can. It's a very effective technique from them to get what they want, because ultimately we need them and we can't actually force them to work.
Good. The police do too much as it is.
Every interaction with the police is a dice roll to see if someone lives or dies.
They could just buy insurance. You know, like doctors, lawyers, and a wide variety of other professionals that deal with liabilities in their field.
Regardless, the police get sued all the time anyways. It's just that the burden currently falls on the taxpayers.