Isn't the reason to cut off a hung start to save the starter not the engine?
Not at all. The starter is insignificant in comparison to the engine in terms of cost or importance. The starter is just an accessory.
Starters have duty limitations which prevent excess use or heat. These duty limitations are provided in terms of start cycles or attempts, and time. Electric starters will have duty cycles expressed in time on and off for cooling, such as one minute on, one off, one on, one off, one on, fifteen off, etc. These are established to provide maximum starter engagement times before cooling is allowed, and the specific cooling periods.
Air driven starters have similiar limitations, usually provided as maximum start attempts, and a max starter time...generally about five minutes to fifteen minutes.
A hung start isn't hard on the starter. It's hard on internal engine components, primarily the burner can.
Part of the procedure following a hung start is generally to cut off fuel to the engine, and to continue motoring the engine to clear any fuel, fire,or vapors that may be there. If the issue were to protect the starter, then certainly we wouldn't keep motoring the engine with the starter.
In fact, if one were to place an interest in the starter rather than the engine, and didn't motor the engine, one could see significant engine damage as a result. Engine damage that may not be confined to just the burner can, but may spread in both directions (toward the diffuser and compressor, as well as the burner cans) as airflow stops with starter disengagement.
Starter use throughout the hung start process is very important. A hung start may result from the introduction of fuel at too low an engine RPM during the start procedure...which may be the result of a low battery or weak starter, but in any event, the procedure for handling a hung start isn't to save the starter. It's to save the engine.
When a hung start occurs, the engine cannot accelerate; it's stuck in a condition of stasis with out the ability to increase RPM. Ideally fuel is being scheduled for that speed, but fuel may continue to feed the engine at an increasing rate, causing other problems...while the RPM doesn't increase. Hung starts can result from a variety of problems, most commonly a failure of an acceleration bleed to operate properly (also for a variety of reasons)...but the central problem is that the engine RPM isn't increasing. There's only one thing to do, and that's shut it down by removing the fuel (and in most cases the ignition, simultaneously). In all cases, you want to keep motoring that engine with the starter...clear evidence that the shutdown isn't anything at all to do with protecting the starter, but the engine.
Every moment that engine stays hung is one more moment for internal components to cook and damage to occur.
When I experience a hung start, I am required to continue to motor the engine for at least 30 seconds. I am also required to motor it until I see a maximum temperature indicated, in my particular case either 100 degrees C or 180 degrees C, depending on which aircraft I'm flying. If no overtemperature has occured, I can continue motoring, reintroduce fuel, and proceed with the second start attempt. I am allowed two start attempts. This occured several days ago during an engine start in Liege. A simple hung start, fuel chop, and reintroduce fuel. It worked fine the second time. At that point, we stopped to review the hung start checklist, and then proceeded to start the other engines.
Motoring to max motoring speed is a good thing for the engine, won't hurt the starter, and produces cooler starts, better engine start protection, longer engine life,and less start abnormal or emergency mishaps.