Why does spin recovery involve cutting the throttle?
Applying engine power can indeed help to end a spin. But the added thrust will make an overspeed in the ensuing spin recovery much more likely. Also, applying asymmetric thrust on the wrong side will flatten and stabilize a spin. These are the main reasons why the engine should be idle in spin recoveries.
Remember the Boeing 307? The first passenger airliner with a pressurized cabin and the civilian version of the B-17. It was test-flown by KLM personell on March 18, 1939. It entered a spin after a stall and the rudder became stuck in spin direction. The pilot applied asymmetric power in order to stop the yawing motion and succeeded in doing so, but in the subsequent recovery the aircraft oversped and the outer wings broke off. Here is the official accident report.

Boeing 307 in flight (picture source). Note the small fixed vertical surface which was replaced after the crash with a much bigger one.
Symmetrical thrust can be helpful or adverse for spin recovery, depending on circumstances. NASA TN D-6575 from December 1971 says that

no systematic study of the effects of applying power during a spin

is known and

The effect of applying symmetrical power in a spin is believed to be insignificant. A number of varied observations have been made through the years by many people, and the conclusions regarding power effects on spins range from favorable to adverse. […] In almost all cases, the results were not obtained under controlled conditions. Therefore, the type of spin, the center of gravity, the angle of attack, the spin rate, and the line of thrust with respect to the center of gravity were not identified.

Therefore, a general recommendation for applying symmetric thrust for spin recovery cannot be made and only asymmetric thrust has shown to be helpful if applied correctly:

the application of thrust had no effect unless the thrust axis was displaced from the center of gravity and thereby produced a moment.
[…]
For asymmetric power for a twin-engine configuration with the engines mounted on the wings, power from only one engine can produce a large asymmetric yawing moment, which will be favorable or adverse to the spin and recovery, depending on the direction of the moment. Both model and full-scale spin-test results of multiengine airplane designs have shown that power on the outboard engine (e.g., the right engine in a left
spin) can create a large prospin yawing moment, which can cause a flatter and faster spin. On the other hand, power on the inboard engine can create an antispin moment to aid spin recovery. Normally, the manipulation of thrust can be confusing and disastrous if the power is applied to the wrong engine. Therefore, unless asymmetric power is necessary to aid recovery, it is generally recommended that for a multiengine airplane, the throttle be retarded to the idle position on all engines during a spin.

It should also be mentioned that the extreme angle of attack at the inboard engine will produce an extreme sideways displacement of the thrust vector on a propeller or uneven flow in a jet intake. In both cases such a flow condition was most likely not anticipated during design and could result in the loss of the propeller rsp. a compressor stall in the jet engine if full thrust is applied.
While I do not know of a propeller having been lost in a spin, during the flight test of the Dornier 328 an extreme sideslip did indeed overload the left propeller, parts of which tore a massive hole into the fuselage.