Why cannot one find the zero in the delta rule for sigmoid? (No closed form to find weights in one-layer perceptron neural network?)

I know that finding the weights of a neural network requires gradient descent as there is no closed form available. I know this from the books, and not knowing exactly why the derivative w.r.t. the weights is not zero-able led me to try to do it. Let's consider the traditional sigmoid MLP, with just one layer and just one datapoint $ \mathbf{x},t $. The gradient vector of the MSE loss function w.r.t. the weights is: Now, how to solve (finding the zero) of the gradient expression? What I could do is to analyze the various factors and see where they individually zero.