Define the variables $$\eqalign{ Y &= {\rm Diag}(y), \quad p = \frac{\partial L}{\partial y}, \quad s = Yp\cr }$$
Then $$\eqalign{ \frac{\partial y}{\partial x} &= Y - yy^T \cr }$$$$\eqalign{ \frac{\partial y}{\partial x} &= \big(Y - yy^T\big) = \big(I-y1^T\big)Y \cr }$$ and $$\eqalign{ \frac{\partial L}{\partial x} &= (Y - yy^T)p \cr &= Yp - (y^Tp)y \cr &= s - (1^Ts)y \cr }$$$$\eqalign{ \frac{\partial L}{\partial x} &= \frac{\partial y}{\partial x}\,\,\frac{\partial L}{\partial y}\cr &= \big(I-y1^T\big)Yp \cr &= \big(I-y1^T\big)s \cr &= s - \big(1^Ts\big)y \cr }$$ which matches your result.