The ncatlab says:

Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left Bousfield localization of $C$ is the left Bousfield localization of $\mathrm{Ho}(C)$.

I wholeheartedly adgree. Does anyone know of results in this direction?

The ncatlab says:

Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left Bousfield localization of $C$ is the left Bousfield localization of $\mathrm{Ho}(C)$.

I wholeheartedly agree. Does anyone know of results in this direction?