For prediction problems, why cant we simply use softmax as activation for hidden layers and no activation function for output layer I am a beginner who is learning how to use neural networks for prediction problems, and almost all tutorials I have found use relu for hidden layer and sigmoid function for output later. My question is, if we simply use softmax for hidden layer and no activation function for output layer, can we get the same result? My inspiration for this is a neural network tutorial by statquest that can be found here https://www.youtube.com/watch?v=CqOfi41LfDw&t=1s. Please keep it simple. I am neither a statistician nor a data scientist

You can, and there is a sense in which doing so would be analogous to a linear probability model after layers of feature extraction, much as using a sigmoid activation function on the output neuron is analogies to logistic regression after layers of feature extraction. In fact, you don’t even have to limit yourself to sigmoid activation in the hidden layers to be analogous to a linear probability model.
A common criticism of a linear probability model is that the output is not guaranteed to be restricted to $[0,1]$, and your neural network that has all sigmoid activation functions in the hidden layers and then no activation function in the final neuron can suffer from this same issue. If this seems like a dealbreaker, then you might not like a model like you’ve proposed that forces the output to be restricted to $[0,1]$, as the sigmoid activation functions in the hidden layers need not force the final output to be restricted to $[0,1]$, even though the values in the hidden neurons will have that restriction.
Cross-entropy (XEnt) loss is common for these classification neural networks. To be analogous to a linear probability model, square loss would be used. Further, XEnt presents problems with predictions outside of the interval, $[0,1]$, due to the $\log(\hat y)$ and $\log(1-\hat y)$ terms (where $\hat y$ is the prediction).
