Climate models simulate a large spread in the projected weakening of the Atlantic meridional overturning circulation (AMOC) over the 21st century. Here, we demonstrate that this uncertainty can be substantially reduced by using a thermal-wind expression that relates the AMOC strength to the meridional density difference and the overturning depth in the Atlantic basin. This expression captures the intermodel spread in AMOC weakening across climate models, with the majority of the intermodel spread arising from overturning depth changes. The overturning depth also establishes a crucial link between the present-day and future AMOC strength. Climate models with a deeper present-day overturning tend to predict greater shoaling under warming. This occurs because their present-day North Atlantic is less stratified, allowing for a deeper penetration of surface buoyancy flux changes, greater density changes at depth, and, consequently, greater AMOC weakening. By integrating observational constraints, we conclude that, regardless of the emission scenario, the AMOC will only experience modest weakening of about 4 Sv by the end of this century. These results indicate that the uncertainty in 21st-century AMOC weakening, and a propensity to predict strong AMOC weakening, can be primarily attributed to climate model biases in accurately simulating the present-day ocean stratification.