In many continuous models, like waiting for a car, we always assume the waiting time $t$ to have an exponential distribution. Why is such an assumption appropriate?
My answer is that we don't. Assuming an exponential arrival time assumes the number of arrivals by time $t$ follows a Poisson process. It is just one of many possible rival time distributions and corresponding point processes that could be used. It is like saying "why do we 'always' assume a normal distribution for continuous random variables?" There too the answer is that we don't. In both cases the method is simple and convenient and there is a limit theorem that sometimes justifies its use.
Recognize that assuming exponential waiting times implies lack of memory. The lack of memory property states that if you are waiting for a bus or car to arrive, and have already waited five minutes, the remaining waiting time has the same exponential distribution that you had when you had just started waiting. This is not always a good assumption. Exponential waiting time/Poisson process are justified under some assumptions of the rarity of events over short time intervals, just like the normal distribution is justified for averages or sums of several observations from some population distribution.