The formal definition of a fine moduli space involves a representable functor(the detailed definition is not quoted here for simplicity).
According to it, in the most general case, just any scheme can be a moduli space, because it represents its own functor of points. But when one gets down into the actual practice of things, it does not seem to be quite the case. Somehow all the moduli spaces studied are taken from the Hilbert scheme. Even an a-priori unrelated thing, like the Jacobian variety, can be constructed from the Hilbert scheme. Why is this? Hilbert scheme is admittedly a very general object; but is that enough reason for the restriction? Aren't there any moduli spaces at all outside group-quotients of subschemes of Hilbert schemes?
There is an important moduli scheme that generalizes the Hilbert scheme called Grothendieck's Quot scheme. It parameterizes quotients. Here we fix an $S$-scheme $X \to S$ and a coherent sheaf $E$ on $X$. Then $$\mathfrak{Quot}_{E/X/S} : \textbf{Sch}/S \to \textbf{Set}$$ maps $T \to S$ the set of quotients of the pulled back sheaf $E_T \in \text{Sh}(X\times_ST)$.
In the event of $E =\mathcal{O}_X$, $$\mathfrak{Quot}_{\mathcal{O}_X/X/S} = \mathfrak{hilb}_{X/S}.$$ (Intuition for this: given a quotient $q : \mathcal{O}_X \to \mathcal{F}$, its kernel is the ideal sheaf of a closed subscheme.)
The (coarse) moduli space of semistable sheaves on a smooth projective variety is constructed as the quotient of an open subscheme of a Quot scheme. And moduli spaces of sheaves are a contemporary topic.
Basically, its because semistability can be expressed using quotients. One takes a sheaf $F$ and for a large enough $m$ we have a surjection $\varphi : \mathcal{O}_X(-m)^{h^0(F(m))} \to F$. A point $\varphi$ in the Quot scheme is of that form if and only if $F$ is semistable and $\rho$ induces an isomorphism $k^{P(F,m)} \to H^0(F,m)$, where $P(F,m)$ denotes the Hilbert polynomial of $F$. So we take the open subscheme of such quotients and mod out by the ambiguity of choice in a basis for $H^0(F,m)$.
To read about Quot schemes in general, I suggest Nitin Nitsure's essay "Construction of Hilbert and Quot Schemes" published in "Fundamental Algebraic Geometry" (a collection of essays, printed by the AMS).
To read about moduli of sheaves, I recommend the "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn (their preliminary section on Quot schemes is good too).
Everything I have said is said (more eloquently) in those references.
And lastly, I do take your point that most moduli spaces we care about are Quot or Hilbert schemes. But, the solutions to many moduli problems we care about, like moduli of stable maps or of complexes of sheaves, are not spaces but rather algebraic spaces, algebraic stacks, or higher stacks. And those require the functorial formalism.