Why do people use CrossEntropyLoss and not just a softmax probability as the loss?
Let me explain it with logistic regression first: 


The output of logistic regression is $\hat{y}= \dfrac{1}{1 + e^{-\theta x}}$
Consider the minimum square loss function $\mathcal{L} = (1 -\hat{y})^2$
substitute $\hat{y}$ in the second equation $\mathcal{L} = (\dfrac{e^\theta x}{1 + e^{-\theta x}})^2$


Tha above formula is not convex, so optimization is much more difficult since the gradient may point to local optima.

Whereas cross entropy looks like.$$\mathcal{L} = -y\log \hat{y} - (1-y)\log (1-\hat{y}) $$ which has two advantages. 


It is convex when subtitute $\hat{y}$ for its value (considering that $y$'s value is either 1 or 0)
For each row in your dataset you are taking on count both labels at the same time


When we generalize from binari logistic regression to multiclass classification, we use the same ideas of using convex loss functions with respect to the output and also loss functions which minimize at the same time for all labels.