In the examples you've given, no. But let's say those firefighters got the address wrong, or are responding to a less-than-truthful 911 call made by someone else, then seemingly yes. Or paramedics slice off your clothing because you were sleeping on a park bench and they imagined you might need medical intervention.
I'm talking about the entity running the services being responsible, and not the individual firefighters/paramedics/cops/etc, assuming they are following department policies in good faith and all that. These things are true costs of providing the services, and should be correctly attributed to avoid perverse incentives. Should we allow firefighters to roll up to a gas station, fill up with diesel, and then declare it's just part of responding to an emergency and not pay?
Of course, examples involving firefighters and paramedics are somewhat limited, as their actions are generally helping rather than interpersonal aggression. In the original example, there is the immediate response of breaking up the fight, which I don't take issue with (assuming it's not a staged fight on a movie set). The victim of the attack being arrested/detained while the police sort things out is where the anti-justice starts to happen - from the victim's informed perspective it is clearly wrong. Only from a limited-information perspective is it necessary. The authorities have limited information at the time, and thus choosing that course is reasonable. But that choice still causes harm that is apparent after all the details are known. And post facto it's not reasonable for authorities to shrug off their responsibility for that.
Furthermore the situation is fraught with perverse incentives and properly assigning responsibility is necessary to minimize them. For example, cops casually resenting the victim for some interpersonal reason and keeping them in jail overnight as extra-judicial punishment.