Lipos are no different than other batteries when it comes to HOW it supplies power.

Batteries, including lipos, do not push, shove, or force power to your system. They are only a source of power that your system can draw from.

Think of your Lipo like a gas tank in your car.
MAHs are the size of the tank.
Discharge c rating is the size of the fuel line to the carburetor.

Your system only draws what it needs from the battery. The battery does not push power into your system. In the car analogy, it doesn't matter how much fuel the tank is holding, the engine will only draw what needs to run. As you press the accelerator the engine will draw more fuel. If the fuel line ( c rating ) is larger than it needs to be it makes no difference to the engine as long as it can supply what the engine wants.

If the fuel line ( c rating ) is to small the engine cannot get enough fuel to run properly.

As to your question weather too small a discharge c rating will damage your system?. I have never heard of that happening. If anything it can damage the battery, because it can't keep up with the demand.