For a lot of things, you can work with a generic time-independent $J_0$ as you suggest; for instance, Audien & Damien work in this context in their book (so for most of the "fundamental" constructions in Hamiltonian Floer theory, you don't need time-dependence in your almost complex structure).
But for other things, it doesn't seem to be easy to do this; for instance, it's often convenient (see for instance Seidel's paper $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings for real development of this idea) to be able to have an action of a Hamiltonian loop $(g_t)_{t \in S^1} \subset Ham(M,\omega)$ on Floer complexes $CF_*(H,J)$ which is essentially an isomorphism of filtered chain complexes (up to a potential shift in action and degree in non-aspherical manifolds), and the natural way to do this in the aspherical case (see either the Seidel paper or the section on the Seidel representation in the chapter on Floer Homology in McDuff & Salamon's J-holomorphic curves and symplectic topology to see how to define the map on capped orbits) is to send $1$-periodic orbits of $\phi_H:=(\phi_H^t)_{t \in [0,1]}$ to $1$-periodic orbits of $g^{-1} \circ \phi_H = (g^{-1}_t \circ \phi_H^t)_{t \in [0,1]}$ (this is the Hamiltonian isotopy generated by $g^*H$), and to then note that a cylinder $u$ satisfies the $(H,J)$-Floer equation if and only if the cylinder $v(s,t)=g^{-1}_t(u(s,t))$ satisfies the $(g^*H,g^*J)$-Floer equation. This shows that the aforementioned map gives an isomorphism of chain complexes between $CF_*(H,J)$ and $CF_*(g^*H,g^*J)$, but for this latter complex to even make sense, you'll need to allow time-dependent almost-complex structures, even if your initial $J$ was time-independent!
So you actually gain something by working with time-dependent almost complex structures. Of course, you pay a bit of a cost too, in that when holomorphic spheres bubble off you need to keep track of the $t$-coordinate in the domain at which the bubble occurs, since the resulting sphere will be $J_t$-holomorphic for that value of $t$ (in the time-independent case you always get a $J_0$-holomorphic sphere, of course). As a consequence, if you don't want to use virtual techniques, then you're forced to work in the setting of strong semipositive symplectic manifolds rather than semipositive symplectic manifolds tout court in order to compensate for the extra dimension of freedom you're adding in permitting time-dependence (the proof of the invariance of Floer homology under changes of the almost complex structure requires that $J$-holomorphic spheres with negative Chern number don't appear in generic $3$-parameter families of almost complex structures. Hofer-Salamon's Floer homology and Novikov rings is a good place for most of the details on this, and it makes clear why you'd need the strong semipositivity hypothesis in the time-dependent case, although they work in the time-independent setting). Also, it stops being so clear that you can just identify the Floer complex of a pair $(f,J)$, where $f$ is a small Morse function, with the Morse complex (with quantum coefficients) of $(f,g_J)$, since autonomous almost complex structures aren't generic in the space of $t$-dependent almost complex structures, so you either have to work with the "equivariant action functional" and quotient out by the additional $S^1$-symmetry here (this is what Floer-Hofer-Salamon do in their "Transversality" paper), or you have to develop the PSS-isomorphism in order to instantiate the usual identification of Floer homology with quantum homology. If I had to guess, I'd imagine that it's this last difficulty which motivated Audin & Damien to do things in the time-independent setting.
So, in some sense, there's a real difference between the two settings, and in practice it's often just more flexible to work with $t$-dependent almost-complex structures, but also, a lot of these differences are mainly just technical and all of the "main results" work in both settings --- and most of these with only fairly minor variations in statements and proofs, but the identification of Floer homology with quantum homology becomes more involved in the $t$-dependent case.
