Is there any principled reason for doing softmax and cross entropy for the loss in for example transformers, rather than doing L2 over the target embeddings and the output from the model?

When the output from your model by necessity is a dot product such as in shallow models I understand why you need to do cross entropy loss. But for models such as rnns and some variants of transformers wouldn’t L2 loss directly on the desired embedding and output work as well or better?