And so vanishing gradients are not a thing anymore?
Good question. That was an issue with tanh as activation function, and before residual connections and normalization layers. Tanh as a normalization but with other activations and residual present apparently is ok.
Batch norm and others are important for faster convergence due to forcing the model to focus creating second and higher order nonlinearities, as a simple shift in mean/std is normalized out, and thus the gradient does not point in a direction that would only change those properties of the output distribution.