Is the Cross Entropy Loss important at all, because at Backpropagation only the Softmax probability and the one hot vector are relevant?
During backpropagation we take the derivative of loss function with respect to all the weight parameters. For cross entropy loss this derivative is simply the difference between output probability and predicted probability. Upto this you're correct.

But you missed what loss function represents. The loss function represents the surface manifold of the network. The purpose of learning is to find the minima of this loss function. 

We use CE loss because the surface define by it is smooth and continuous, hence we can find the derivative easily. 

Mathematically that derivative comes out to be a simple difference, but theoretically we're differentiating the surface defined by the loss function, which is important.