There is an error with the presented variance of the MLE $\hat{\lambda}=\overline{X}$. The correct expression would be: $\hat{SE}(\hat{\lambda}) = \sqrt{1/I_n(\hat{\lambda})}=\sqrt{\frac{\overline{X}}{n}}$.
This implies changes over the Wald statistic and the simulation, of course.
There is an error with the presented variance of the MLE $\hat{\lambda}=\overline{X}$. The correct expression would be: $\hat{SE}(\hat{\lambda}) = \sqrt{1/I_n(\hat{\lambda})}=\sqrt{\frac{\overline{X}}{n}}$.
This implies changes over the Wald statistic and the simulation, of course.