Te reason you want the decaying material is so it can be food for the "good" live bacteria. If it was live the bacteria would not feed on it and you wouldnt be cycling your tank.
I think you are misunderstanding the concept of a tank cycle, and you have some research to do.
The reason why people use fish is because they eat food a produce waste. It usually takes much longer to cycle. The ammonia does not spike as high, you wind up with less good bacteria leading your tank to mini cycle every time you increase the amount of waste in the tank by adding waste by fish and food.
In order to cycle your tank with a fish people use the most hardy species. The problem here is not only is it extremely inhumane to subject your fish to a tank cycle but also generally speaking the more hardy the fish the more aggressive. Once the fish is subjected to the cycle and if perhaps it survives it will be extremely aggressive towards any new tank mates.
You need flow. If you let decaying material sit in the tank with no flow the water will get stagnant and produce scent. If you have flow the material will move, dissipate, and if you have surface agitation you will introduce o2 into the water column and not smell. The flow will also circulate nutrients from the dead shrimp to other parts of the tank increasing nitrifying bacteria growth.
Freshwater tanks can not have decaying material because they do not have any real means of removing it unless you have a great filtration system. The material raises the ammonia levels and stresses the inhabitants. In marine tanks we have biological filtration, flow, skimming ect...All aid in removing excess waste and ammonia....
Do some more research.
Do you have rock in the tank? Filling the tank with rotting waste without any way of housing bacteria wouldn't be smart.
Its is very smart like I said previously to introduce established water, sand, crushed coral, rock....from a very established tank from a friend or LFS a cup is really all you need of each but the more the better. IT will populate your tank much faster.