VNL calculates the Young's modulus as Y_i = 1/S_ii (i=x,y,z) where S is the inverse of the elastic constants matrix.
We do not calculate the bulk modulus and the shear modulus directly, but from the elastic constants, as well. Since there are several conventions how to do this exactly (Hill, Voigt, Reuss) which all use slightly different formulas, it is more straightforward to calculate the Young's modulus from the elastic compliance matrix, since that does not rely on any of these conventions.