The inverse relationship between the saturation magnetic flux density (Bs) and coercivity (Hc) of Fe-based amorphous alloys is a very active research topic that has been extensively debated. In this work, we conducted a detailed investigation on the magnetic softness of Fe83.2-xCoxB10C6Cu0.8 (x = 0 and 6 at.%) amorphous alloys based on analysis of the surface morphology, microstructure, magnetic anisotropy, and magnetic domain structure. Enhanced magnetic softness-magnetization synergy was realized in the present alloys by magnetic field annealing (MFA) during the de-stressing process. A dramatic 84 % reduction of Hc to 2.2 A/m was achieved for the Co-doped alloy under MFA, exhibiting excellent magnetic performance with a superb Bs of 1.86 T. The consistency between the experimental results and theoretical analysis revealed that the MFA process can mitigate the trade-off between stress-induced anisotropy and induced uniaxial anisotropy owing to the homogenized structure formed by field annealing. Thus, the process favored a low Hc due to the significant continuous decline in the total magnetic anisotropy, which coincided well with the results of Magneto-optical Kerr microscopy. The study elucidates a mechanism for tuning Hc in Co-doped alloy systems and affords a possible pathway for softening amorphous alloys with high Bs.