So be it. I was a firefighter/paramedic. One of my state's laws about operating emergency vehicle in "emergency mode" (i.e. lights and sirens) is that the vehicle operator is permitted to disobey any and all road laws[1] provided they are doing so with due regard and with a presumption of fault in the event of any incident, i.e. you are default assumed to be at fault unless demonstrable otherwise. Such liability can transcend department or agency into personal liability against the operator.

How is that hedged against, in practice? Most departments have their SOPs for emergency mode driving, for example mine said "You can exceed the speed limit by no more than 20mph, and subtract 5mph for any confounding condition, such as fog, rain, nighttime without streetlighting" and "You must come to a stop or to a sufficiently slow speed that you can affirmatively clear your passage through an intersection without incident." Stay within those guidelines and the department and their insurer agrees to indemnify your personal liability. Outside of those, you're on your own.

That's tangential. I have no problem saying "police pension funds are responsible for these compensation claims". Then the fund itself can decide whether they want to police themselves better, seek insurance coverage at their expense, or (ideally!) both.