Yes, I think the use of the exponential distribution in the Mathpages webpage is just some sort of 'academic exercise'. He's working with the simpler, more elegant distribution because it's easier than using the actual one.

The actual distribution for age at death for humans follows the Gompertz Law, which was devised in 1825. It consists of an exponential raised to the power of another exponential. This blog post gives some details of it:

The blog post also compares the double exponential distribution with the ordinary exponential distribution (the latter referred to as the 'lightning bolt theory' in the post).