Magnetic vector potential and displacement current
The expression for the vector potential as an integral over the current is derived for electrostatic fields, or for fields slowly varying enough to neglect the displacement current. So your observation is correct that these expressions neglect the dependence of B upon displacement current.
Edit, after dusting off my EM textbook:
Although the equation for A is often derived for the quasi-static case, by using retarded time in the equation, it turns out, we end up including the dependence of A on the displacement current in the non-quasistatic case. This occurs after a lot of mathematical detail, going into the Coulomb Gauge and breaking the current into transverse (zero divergence) and and irrotational (zero curl) components. The contributions from the longitudinal current term ends up canceling with a term from the scalar potential, leaving you with the wave equation for A, with the transverse component of J as the source. The solution to this is your equation for A with retarded time. When you do this, the displacement current contribution ends up being built into the equation, for which the solution is your equation for A using the retarded time. I believe this implies that the current distribution must have zero divergence.
To me, like you, it is not at all apparent looking at the equation that the displacement current contribution to A is baked into the equation for the vector potential (with retarded time), but it ends up being true.