Fundamentally H' is incorrect. The ground state of the harmonic oscillator will still have kinetic and potential energy and the sum of these energies will still be $\hbar \omega/2$. These energies contribute to inertial and gravitational mass, but this would be practically impossible to demonstrate experimentally. Also, the zero point energy gives rise to the Casimir Polder effect. This effect is real and needs to be taken into account in the design of MEMSs (micro-electromechanical systems).
In QFT an infinite continuum of degrees of freedom is postulated, to be compared with a finite number of DoF in condensed matter. This causes a quasi infinite vacuum energy, only limited if the universe is indeed finite. This infinite ZPE can of course not be real. One way to solve this is to use H', but then you lose very real physics. The alternative is to consider only those harmonic oscillators that correspond to actual system DoF, as in condensed matter physics.
In cases where ZPE is irrelevant use H' by als means.