Also, heavy urban traffic where some health issues usually occur is not the easiest stage for racing tricks...
Then, engines that provide a better low-end torque and have less electric features prone to fail and also lead to some performance issues while using some high energy-demanding devices such as a defibrilator are desirable. And the best answer for these requirements is a Diesel engine.
Tougher than a gasser with a similar performance, Diesels are a great option to use in off-road-capable rescue vehicles due to their increased reliability due to the absence of an electric ignition setup, without spark plugs, ignition cables, coils and specially the distributor that is widely sensible to harsh environmental conditions. But even a simple coil can be a source of trouble, I still remember an occasion when I had some problems with the coil of a 1.0L Opel Corsa after getting the engine washed...
For some operators such as military corps, the dependability of a Diesel is a #1 priority, for example...
Also, another advantage is the adaptability to the use of alternative fuels, going from biodiesel (which biggest consumer in the world is currently U.S. Navy) and ethanol to waste cooking oil or some lube oils from different grades. Due to the compression-ignition setup relying on the injection timing, there are no pre-ignition issues like an Otto (average spark-ignited 4-stroke) running on low-octane gasoline, so no sensible power/torque losses.
Not just for military tactical use, the possibilities of biofuels are a good alternative for regular civilian ambulances due to their lighter environmental impact...While they're often quoted as "agricultural tools" due to the noise, vibration and harshness (NVH) from some older developments, in the last 2 decades their evolution has been noticeable, and the ride on a recent Diesel-powered vehicle can be as comfortable as in a gasser. Improvements on electronic injection management systems apart, the evolution in the turbocharging has benefitted the Diesel in the performance field, and also due to its already stronger-than-gasser construction these engines don't sacrifice too much reliability under some heavy boost. Nowadays, some smaller-displacement Diesels are spotting some performance levels that once were restricted to larger gasoline-powered engines.
As far as vibration goes, counterbalancing shafts and hydraulic motor mount cushions have been a successful way to deal with this issue. Noise is attenuated with the use of some sound-deadening materials in the linings, and also considering the sirens are usually louder than a Diesel engine...
Another usual report against Diesels is related to cold-start capability. Newer engines fitted with direct injection have quicker start-ups in cold weather than some earlier indirect-injection ones, but some features like glow-plugs and grid-heaters have been still widely used to ease the cold-start...
So, even with some misconceptions around, Diesel engines are a good feature for ambulances...