Supplementing other comments and @JimHumphreys' answer: Thinking of automorphic forms as living only on quotients of symmetric spaces or of real Lie groups leaves one with an extremely awkward neo-classical version of Hecke operators, and, more pointedly, no action in sight of the corresponding $p$-adic groups, so no way to take advantage of what is known about their representation theory. To "convert" automorphic forms as functions on something like $G(\mathbb Z)\backslash G(\mathbb R)$ to automorphic forms on $G(\mathbb Z[1/p])\backslash G(\mathbb R)\times G(\mathbb Q_p)$, which not only makes the $p$-Hecke operators much more tractable, but, in fact, happily, allows the direct application of the representation theory of $G(\mathbb Q_p)$.
(One main virtue of the latter is the wonderful Borel-Casselman-Matsumoto theorem, that shows that not only spherical representations, but admissible repns with Iwahori-fixed vectors, are subrepresentations (and quotients) of unramified principal series. This also does account for the "square-free level" condition of many classical papers on modular forms, since these exactly correspond to Iwahori-fixed vectors...)
For that matter, the physical space $G(\mathbb Z[1/p])\backslash G(\mathbb R)\times G(\mathbb Q_p)$ arises very reasonably, as a sort of non-abelian solenoid, namely, the (projective) limit of $\Gamma(p^n)\backslash G(\backslash R)$, as $\Gamma(p^n)$ runs over principal $p$-power congruence subgroups. One would find that the limitands in this limit arise inevitably in looking at $p$-power Hecke operators even at level one.