Let's say we have a random variable that can only take positive values (time until the next bus arrives for example). An obvious choice to me is to take the PDF of the normal distribution but only on positive values of $x$.
$$f_X(x) = \begin{cases} \sqrt{\frac{2}{\pi \sigma^2}}e^{-\frac{x^2}{4 \sigma^2}}, \;\;x>0\\ 0, \;\; x \leq 0 \end{cases}$$
I haven't seen this distribution being used, however. It's a special case of the truncated normal. Is there a reason this doesn't get the same consideration as distributions like the exponential, log normal, log logistic and other distributions defined over positive real numbers?
It is not so obvious in the case you took as example. In particular, the most common distribution for random arrivals is a poisson and, as known, interarrival (time between two arrivals) is clearly a Negative exponential
In other situation, also your truncated gaussian is used. In some other situation, also a (non truncated) Gaussian can be used. This because, even if its domain is $\mathbb{R}$, actually it is in $(\mu-3\sigma;\mu+3\sigma)$ thus, for example, if you have to represent the random diameter of a bolt's hole you can use a Gaussian, ie
$$X\sim N(10;0.1)$$
actually you will not have negative values being
$$P(X<0)\approx 8.98\cdot 10^{-220}$$