Do we care about multiple zeta functions?

It may help clarify things to work out a specific example, although the OP may know this. In case $n=3$, the double Dirichlet series evaluates as $$ \sum_{m,n=1}^{\infty} \frac{A_F(m,n)}{m^{w} n^s} = \frac{L(\overline{F},w) L(F, s)}{\zeta(s+w)}.$$ Here $\overline{F}$ is the contragredient of $F$. This is known as Bump's double Dirichlet series, and this is worked out in Section 6.6 of Goldfeld's book, Automorphic forms and $L$-functions for the group $GL(n,\mathbb{R})$. There are references there for generalizations.

It seems clear (to me) that the $GL_3$ automorphic $L$-function $L(F,s)$ (and its contragredient, and $\zeta$) are the fundamental objects, but also that there are many interesting Dirichlet series that one can construct from an automorphic form.

To address the conceptual question, the $L$-function essentially characterizes the automorphic representation and can be studied locally (associating local $L$-functions to local components of the global representation), so it seems to me there is little need (a priori) for using a more complicated Dirichlet series to study these representations.

On the other hand, multiple Dirichlet series typically do not have Euler products, so do not admit a local-global study, at least in a naive way (though see Bump's survey article). Of course, as Matt indicates, they are useful for studying $L$-functions.

Moreover, from an arithmetic point of view, $L$-functions are naturally related to varieties such as elliptic curves. As far as I know (though I am not an expert on multiple Dirichlet series), there is no direct connection between MDS and counting points on varieties.