Why do we multiply two wavefunctions in valence bond theory instead of adding them?
Actually, the MO theory wavefunction can also be written as a product. While the individual molecular orbitals are just weighted sums of the atomic orbitals, the wavefunction can, to a first approximation, be written as a Slater determinant of the occupied orbitals. 

For your example of the hydrogen molecule, this would mean taking the antisymmetrized product of the occupied $\sigma$ spin orbitals. (Note, if you haven't encountered this before, spin orbitals are a spatial and spin description of a single electron, rather than the commonly encountered spatial orbitals which describe two electrons of opposite spin.