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:
But do we know the distribution at age of death from observing deaths? I would have thought it is hard to model - considered as a survival curve it will be at least triphasic, relatively steep for a short time at very young ages, then rising slowly in more or less linear fashion from say 2 years to about 60 years, then after that it gets more interesting.