Why isn't life expectancy normally distributed?
The histogram you show isn't a distribution of age-at-death for individuals, it's a distribution of life expectancies for different countries.  The post you link takes data from a table on the Wikipedia page List of countries by life expectancy.  Interpreting the histogram you posted, we see average life expectancy seems to hit a wall around 85.  This does not mean that individual human life span is not normally distributed.
Excluding deaths at infancy, the peak of the death rate curve appears to be at about 88 years for women in the United States in 2014, as seen in a blog post by the The Max Planck Institute for Demographic Research:


Lifespans are increasing and becoming more similar: While life expectancy of US women rose by almost 19 years between 1933 and 2014 (from 62.8 to 81.3 years), the variance in the distribution of deaths over age shrank by ten years. Data: Human Mortality Database © MPIDR