Most cars are stolen just to be 'joy ridden' - so you're going to need the police to have the ability to pursue and stop them.
The problem you have is once a car is making off from a response police panda car, there are only 2 ways its going to end - it stops of its own accord or it crashes.
Since its making off from police, its obviously not going to stop of its own accord, so the only way its going to end is by crashing. That's when you get traffic cars involved with stingers or T-Pac to give you another tactic. But until traffic can get to the pursuit (depending on where they are travelling from) you're going to need the response panda car to maintain the pursuit.
There are a whole host of other jobs which police will go to where every second counts - ongoing domestics, missing people, suicidal people, fights, burglaries in progress, suspicious deaths etc etc
The reason why ambulances & fire engines travel slower is that they are a lot heavier. Fire engines will have 2-3 tons of water on board and ambulances start life as a short wheel base flat-bed lorry and have all their equipment added.