Why is Poisson distribution's expected value not on the curve's maximum
To have mean = mode you need a symmetric distribution. Try with a poisson with $\lambda =10$ or more and you will have $\text{argmax}(x) =\lambda$
This phenomenon is not particular to a Poisson distribution. It's true whenever the distribution is skewed.
Which of the three "averages" you use in any particular application depends on the context.
