Consider the function $f\colon \mathbb Z \to \mathbb Z$ given by $f(x) = g^x \bmod p$ where $g$ has order $q$ modulo $p$. If $x \equiv y \pmod q$, then necessarily $f(x) = f(y)$, since by hypothesis $x = y + \ell q$ for some $\ell$, so
\begin{equation}
g^x \equiv g^{y + \ell q} \equiv g^y g^{\ell q}
\equiv g^y (g^q)^\ell \equiv g^y 1^\ell \equiv g^y
\pmod p.
\end{equation}
Note that $f$ is merely a function from integers to integers, but this proposition shows that there is additional periodic structure to its input: $f(x) = f(x + \ell q)$ for any $\ell$. So if we consider the coset $\overline x = x + q\mathbb Z = \{\dots, x - 2q, x - q, x, x + q, x + 2q, x + 3q, \dots\}$ of $q \mathbb Z$, $f$ is the same on every element of $\overline x$. Thus we can think of $f$ as being defined on the quotient $\mathbb Z/q\mathbb Z$, the set of all cosets of $q\mathbb Z$, rather than on $\mathbb Z$. More jargomatically: there is a unique natural $\overline f\colon \mathbb Z/q\mathbb Z \to \mathbb Z$ defined in the quotient $\mathbb Z/q\mathbb Z$ so that $f = \overline f \circ \phi_q$, where $\phi_q\colon \mathbb Z \to \mathbb Z/q\mathbb Z$ is the natural projection $\phi_q(x) = x + q\mathbb Z = \overline x$ onto $\mathbb Z/q\mathbb Z$.
In other words, even if we treat $x \mapsto g^x \bmod p$ as a function on integers, we can always break it down into two steps, the first of which is reduction modulo $q$. So, whether you like it or not, if you are computing $g^x$ for integers outside $0 \leq x < q$, the function can still always be defined in terms of doing $x \bmod q$ first.
What's much weirder is not reduction modulo $q$—which as above is necessary if you're going to use the result as an exponent—but the chain of reduction modulo $p$ and then modulo $q$. This was a point of contention in the provable security literature on DSA and ECDSA. You could imagine computing $g^k \bmod q$ directly, but it wouldn't make a functioning signature scheme. What you really want is a uniform random map from $\mathbb Z/p\mathbb Z$ to $\mathbb Z/q\mathbb Z$ which has no properties that an adversary could exploit.
It just happens that $h \mapsto (h \bmod p) \bmod q$ sorta kinda seems to work but as far as I know nobody has a good handle on why even in recent analyses like Fersch–Kiltz–Poettering 2016 (paywall-free), which models it roughly as a random oracle while tidying up nearly everything else about the long and sordid history of attempts to prove DSA and ECDSA security.
