Robustness to disorder is the defining property of any topological state. The ultimate disorder limits to topological protection are still unknown, although a number of theories predict that even in the amorphous state a quantized conductance might yet reemerge. Here we report that in strongly disordered thin films of the topological material Sb$_2$Te$_3$ disorder-induced spin correlations dominate transport of charge—they engender a spin memory phenomenon, generated by the nonequilibrium charge currents controlled by localized spins. We directly detect a glassy yet robust disorder-induced magnetic signal in films free of extrinsic magnetic dopants, which becomes null in a lower-disorder crystalline state. This is where large isotropic negative magnetoresistance (MR)—a hallmark of spin memory—crosses over to positive MR, first with only one e$^2$/h quantum conduction channel, in a weakly antilocalized diffusive transport regime with a 2D scaling characteristic of the topological state. A fresh perspective revealed by our findings is that spin memory effect sets a disorder threshold to the protected topological state. It also points to new possibilities of tuning spin-dependent charge transport by disorder engineering of topological materials