The process of backpropagation is figuring out which input has the greatest effect on output and adjusting the weights on the input to fit the output curve. In other words, assuming a trained single layer neural network, the weights alone will tell you which inputs have the greatest effect on output.

Sorry for the late reply, just saw your comment. I think what you're describing is training a model and then using feature importance. This is also a valid method for calculating sensitivities and it's used if the function from inputs to outputs is unknown. However, in our case, this function is known so there's no need to train a predictive model.