How is zero conditional mean assumption "stronger" than uncorrelated assumption? I am trying to understand what the zero conditional mean assumption ($\mathbb{E}[u\vert X]=0 $) encompasses in addition to a zero-correlation assumption ($\text{Corr}(X,u)=0$). I assume it must be "stronger" (by stronger I mean covers more cases) assumption, as otherwise we could replace it with a zero-correlation assumption. So what would be the cases where the zero-correlation assumption is not sufficient?

The result follows from the law of iterated expectations,
$$
E[X^T \epsilon]=E[X^TE[\epsilon | X ]]=E[X^T0]=E[0]=0.
$$
For examples where the stronger condition is not satisfied but the weaker may be, consider Proving OLS unbiasedness without conditional zero error expectation? or Non Linear Endogeneity.
