If Ampere's law implies the Biot-Savart law, which implies Gauss's law for magnetism, does that mean Maxwell's equations are redundant?

Since there is no magnetic charge term in the Biot-Savart law, it is only correct if Gauss's law for magnetism ($\nabla \cdot \mathbf{B} = 0$) is true and there are no magnetic monopoles. So it makes sense that Gauss's law can be derived from the Biot-Savart law.

However, the Biot-Savart law cannot be derived from the Maxwell-Ampère Law without implicitly assuming Gauss's law. In general, we know this both because of the lack of a magnetic charge term and because as Giorgio pointed out, the curl and divergence of a vector field are independent quantities. The specific problem with the proof you cited is that it assumes that a continuous vector potential $\mathbf{A}$ can be constructed such that $\nabla \times \mathbf{A} = \mathbf{B}$, which is not true if there are magnetic monopoles.

Curl and divergence of a vector field are independent quantities ( Helmholtz's theorem allows to reconstruct a vector field if both are known). So, it is impossible to deduce anything for each of these two quantities from the other.