This is a common approach in some introductory statistics textbooks. The alternative hypothesis can be directional (e.g., $H_a : \mu > \mu_0$) or non-directional (e.g., $H_a : \mu \ne \mu_0$), but the null hypothesis is always written as an equality (e.g., $H_0 : \mu = \mu_0$).
Your evaluation is correct: this would be the mutually exclusive alternative only for the non-directional test. The appropriate mutually exclusive option for the first alternative hypothesis above would properly be $H_0 : \mu \le \mu_0$.
So, ¿why do textbook authors sometimes just always write the null with the equality sign? Well, it comes down to what you can (and cannot) draw. I can draw a picture of a hypothetical world where the population mean is a given value (say $\mu_0$). I can sketch the normal curve, indicate the center is at $\mu_0$, and I'm good to go. What I can't do is draw infinitely many other such curves were $\mu \le \mu_0$.
OK...but ¿won't the $P$-values be different if I drew different curves? Yes, they would, but if you conduct a thought-experiment of what the new $P$-value would be if you did have a normal curve with a shifted mean, that new $P$-value will always be less than the one you calculated with the fixed null hypothesis.
And in the end, technically, I can't calculate a separate $P$-value for the infinite options indicated in $H_0: \mu \le \mu_0$, but I can calculate one for $H_0 : \mu = \mu_0$. (Well, not if we aren't going down a Bayesian path...)**
Hope this helps justify the pedagogic rationale behind this (seemingly) wrong conventional notation.
**This comment is based on the more simplistic definition of $P$-value used in most introductory statistics textbook. A more general definition of the $P$-value can account for this, and is described in another answer below.