Does this imply that any arbitrary time evolution of current density can be defined beforehand, and the corresponding fields always found that satisfy Maxwell's equations?
Yes, given a charge density $\rho(\mathbf r,t)$ and a current density $\mathbf J(\mathbf r,t)$, you can find fields $\mathbf E(\mathbf r,t)$ and $\mathbf B(\mathbf r,t)$ satisfying Maxwell’s equations.
See Wikipedia for the integrals giving the scalar potential $\varphi$ and vector potential $\mathbf A$ that solve the nonhomogeneous wave equations with sources $\rho$ and $\mathbf J$. The fields derived from these potentials will satisfy Maxwell’s equations.
One way to think about this is that an arbitrary charge and current density can be considered a swarm of moving point charges. The fields of an arbitrarily moving point charge is known, based on the Liénard-Wiechert potentials. The fields of the swarm are simply the superposition of the fields of all the point charges, by the linearity of Maxwell’s equations.
ADDENDUM: As @knzhou points out in another answer, the $\rho$ and $\mathbf J$ can’t be completely arbitrary. They have to satisfy the physical constraint of current conservation, $\partial\rho/\partial t+\nabla\cdot\mathbf J=0$.