(Note that in the part you quoted, the statement was conditional; the sentence itself didn't assume exponential survival, it explained a consequence of doing so. Nevertheless assumption of exponential survival are common, so it's worth dealing with the question of "why exponential" and "why not normal" -- since the first is pretty well covered already I'll focus more on the second thing)
Normally distributed survival times don't make sense because they have a non-zero probability of the survival time being negative.
If you then restrict your consideration to normal distributions that have almost no chance of being near zero, you can't model survival data that has a reasonable probability of a short survival time:
Maybe once in a while survival times which have almost no chance of short survival times would be reasonable, but you need distributions that make sense in practice -- usually you observe short and long survival times (and anything in between), with typically a skewed distribution of survival times). An unmodified normal distribution will rarely be useful in practice.
[A truncated normal might more often be a reasonable rough approximation than a normal, but other distributions will often do better.]
The constant-hazard of the exponential is sometimes a reasonable approximation for survival times.. For example, if "random events" like accident are a major contributor to death-rate, exponential survival will work fairly well. (Among animal populations for example, sometimes both predation and disease can act at least roughly like a chance process, leaving something like an exponential as a reasonable first approximation to survival times.)
One additional question related truncated normal: if normal is not appropriate why not normal squared (chi sq with df 1)?
Indeed that might be a little better ... but note that that would correspond to an infinite hazard at 0, so it would only occasionally be useful. While it can model cases with a very high proportion of very short times, it has the converse problem of only being able to model cases with typically much shorter than average survival (25% of survival times are below 10.15% of the mean survival time and half of the survival times are less than 45.5% of the mean; that is median survival is less than half the mean.)
Let's look at a scaled $χ^2_1$ (i.e. a gamma with shape parameter $\frac12$):
[Maybe if you sum two of those $χ^2_1$ variates... or maybe if you considered noncentral $χ^2$ you would get some suitable possibilities. Outside of the exponential, common choices of parametric distributions for survival times include Weibull, lognormal, gamma, log-logistic among many others ... note that the Weibull and the gamma include the exponential as a special case]