Why are survival times assumed to be exponentially distributed?

Exponential distributions are often used to model survival times because they are the simplest distributions that can be used to characterize survival / reliability data. This is because they are memoryless, and thus the hazard function is constant w/r/t time, which makes analysis very simple. This kind of assumption may be valid, for example, for some kinds of electronic components like high-quality integrated circuits. I'm sure you can think of more examples where the effect of time on hazard can safely be assumed to be negligible.

However, you are correct to observe that this would not be an appropriate assumption to make in many cases. Normal distributions can be alright in some situations, though obviously negative survival times are meaningless. For this reason, lognormal distributions are often considered. Other common choices include Weibull, Smallest Extreme Value, Largest Extreme Value, Logistic, etc. A sensible choice for model would be informed by subject-area experience and probability plotting. You can also, of course, consider non-parametric modeling.

A good reference for classical parametric modeling in survival analysis is: William Q. Meeker and Luis A. Escobar (1998). Statistical Methods for Reliability Data, Wiley

To add a bit of mathematical intuition behind how exponents pop up in survival distributions:

The probability density of a survival variable is $f(t) = h(t)S(t)$, where $h(t)$ is the current hazard (risk for a person to "die" this day) and $S(t)$ is the probability that a person survived until $t$. $S(t)$ can be expanded as the probability that a person survived day 1, and survived day 2, ... up to day $t$. Then: $$ P(survived\ day\ t)=1-h(t)$$ $$ P(survived\ days\ 1, 2, ..., t) = (1-h(t))^t$$ With constant and small hazard $\lambda$, we can use: $$ e^{-\lambda} \approx 1-\lambda$$ to approximate $S(t)$ as simply $$ (1-\lambda)^t \approx e^{-\lambda t} $$ , and the probability density is then $$ f(t) = h(t)S(t) = \lambda e^{-\lambda t}$$

Disclaimer: this is in no way an attempt at a proper derivation of the pdf - I just figured this is a neat coincidence, and welcome any comments on why this is correct/incorrect.

EDIT: changed the approximation per advice by @SamT, see comments for discussion.

You'll almost certainly want to look at reliability engineering and predictions for thorough analyses of survival times. Within that, there are a few distributions which get used often:

The Weibull (or "bathtub") distribution is the most complex. It accounts for three types of failure modes, which dominate at different ages: infant mortality (where defective parts break early on), induced failures (where parts break randomly throughout the life of the system), and wear out (where parts break down from use). As used, it has a PDF which looks like "\__/". For some electronics especially, you might hear about "burn in" times, which means those parts have already been operated through the "\" part of the curve, and early failures have been screened out (ideally). Unfortunately, Weibull analysis breaks down fast if your parts aren't homogeneous (including use environment!) or if you are using them at different time scales (e.g. if some parts go directly into use, and other parts go into storage first, the "random failure" rate is going to be significantly different, due to blending two measurements of time (operating hours vs. use hours).

Normal distributions are almost always wrong. Every normal distribution has negative values, no reliability distribution does. They can sometimes be a useful approximation, but the times when that's true, you're almost always looking at a log-normal anyway, so you may as well just use the right distribution. Log-normal distributions are correctly used when you have some sort of wear-out and negligible random failures, and in no other circumstances! Like the Normal distribution, they're flexible enough that you can force them to fit most data; you need to resist that urge and check that the circumstances make sense.

Finally, the exponential distribution is the real workhorse. You often don't know how old parts are (for example, when parts aren't serialized and have different times when they entered into service), so any memory-based distribution is out. Additionally, many parts have a wearout time that is so arbitrarily long that it's either completely dominated by induced failures or outside the useful time-frame of the analysis. So while it may not be as perfect a model as other distributions, it just doesn't care about things which trip them up. If you have an MTTF (population time/failure count), you have an exponential distribution. On top of that, you don't need any physical understanding of your system. You can do exponential estimates just based on observed part MTTFs (assuming a large enough sample), and they come out pretty dang close. It's also resilient to causes: if every other month, someone gets bored and plays croquet with some part until it breaks, exponential accounts for that (it rolls into the MTTF). Exponential is also simple enough that you can do back-of-the-envelope calculations for availability of redundant systems and such, which significantly increases its usefulness.

Some ecology might help answer the "Why" behind this question.

The reason why exponential distribution is used for modeling survival is due to the life strategies involved in organisms living in nature. There's essentially two extremes with regard to survival strategy with some room for the middle ground.

Here's an image that illustrates what I mean (courtesy of Khan Academy):

This graph plots surviving individuals on the Y axis, and "percentage of maximum life expectancy" (a.k.a. approximation of the individual's age) on the X axis.

Type I is humans, which model organisms which have an extreme level of care of their offspring ensuring very low infant mortality. Often these species have very few offspring because each one takes a large amount of the parents time and effort. The majority of what kills Type I organisms is the type of complications that arise in old age. The strategy here is high investment for high payoff in long, productive lives, if at the cost of sheer numbers.

Conversely, Type III is modeled by trees (but could also be plankton, corals, spawning fish, many types of insects, etc) where the parent invests relatively little in each offspring, but produces a ton of them in the hopes that a few will survive. The strategy here is "spray and pray" hoping that while most offspring will be destroyed relatively quickly by predators taking advantage of easy pickings, the few that survive long enough to grow will become increasingly difficult to kill, eventually becoming (practically) impossible to be eaten. All the while these individuals produce huge numbers of offspring hoping that a few will likewise survive to their own age.

Type II is a middling strategy with moderate parental investment for moderate survivability at all ages.

I had an ecology professor who put it this way:

"Type III (trees) is the 'Curve of Hope', because the longer an individual survives, the more likely it becomes that it will continue to survive. Meanwhile Type I (humans) is the 'Curve of Despair', because the longer you live, the more likely it becomes that you will die."

To answer your explicit question, you cannot use the normal distribution for survival because the normal distribution goes to negative infinity, and survival is strictly non-negative. Moreover, I don't think it's true that "survival times are assumed to be exponentially distributed" by anyone in reality.

When survival times are modeled parametrically (i.e., when any named distribution is invoked), the Weibull distribution is the typical starting place. Note that the Weibull has two parameters, shape and scale, and that when shape = 1, the Weibull simplifies to the exponential distribution. A way of thinking about this is that the exponential distribution the simplest possible parametric distribution for survival times, which is why it is often discussed first when survival analysis is being taught. (By analogy, consider that we often begin teaching hypothesis testing by going over the one-sample $z$-test, where we pretend to know the population SD a-priori, and then work up to the $t$-test.)

The exponential distribution assumes that the hazard is always exactly the same, no matter how long a unit has survived (consider the figure in @CaffeineConnoisseur's answer). In contrast, when the shape is $>1$ in the Weibull distribution, it implies that hazards increase the longer you survive (like the 'human curve'); and when it is $<1$, it implies hazards decrease (the 'tree').

Most commonly, survival distributions are complex and not well fit by any named distribution. People typically don't even bother trying to figure out what distribution it might be. That's what makes the Cox proportional hazards model so popular: it is semi-parametric in that the baseline hazard can be left completely unspecified but the rest of the model can be parametric in terms of its relationship to the unspecified baseline.

tl;dr- An expontential distribution is equivalent to assuming that individuals are as likely to die at any given moment as any other.

Derivation

1. Assume that a living individual is as likely to die a moment as at any other moment.

2. So, the death rate $-\frac{\text{d}P}{\text{d}t}$ is proportional to the population, $P$.

$$-\frac{\text{d}P}{\text{d}t}{\space}{\propto}{\space}P$$

3. Solving on WolframAlpha shows:

$$P\left(t\right)={c_1}{e^{-t}}$$

So, the population follows an exponential distribution.

Math note

The above math is a reduction of a first-order ordinary differential equation (ODE). Normally, we would also solve for $c_0$ by noting the boundary condition that population starts at some given value, $P\left(t_0\right)$, at start-time $t_0$.

Then the equation becomes: $$P\left(t\right)={e^{-t}}P\left({t_0}\right).$$

Reality check

The exponential distribution assumes that people in the population tend to die at the same rate over time. In reality, death rates will tend to vary for finite populations.

Coming up with better distributions involves stochastic differential equations. Then, we can't say that there's a constant death likelihood; rather, we have to come up with a distribution for each individual's odds of dying at any given moment, then combine those various possibility trees together for the entire population, then solve that differential equation over time.

I can't recall having seen this done in anything online before, so you probably won't run into it; but, that's the next modeling step if you want to improve upon the exponential distribution.