Why is the atomic mass unit less than the mass of both a neutron and a proton?
The source of the discrepancy is the mass defect, which is described in the article below as :
The difference between the sum of the masses of the components and the measured atomic mass is called the mass defect of the nucleus. Just as a molecule is more stable than its isolated atoms, a nucleus is more stable (lower in energy) than its isolated components. Consequently, when isolated nucleons assemble into a stable nucleus, energy is released. According to Equation 4, this release of energy must be accompanied by a decrease in the mass of the nucleus.
The equation four referenced is just E=mc^2.  When these nucleons form carbon-12, which is the reference isotope for the definition of the amu, the mass defect appears, as you mentioned, by conversion to binding energy.

https://chem.libretexts.org/Courses/Grand_Rapids_Community_College/CHM_120_-_Survey_of_General_Chemistry/2%3A_Atomic_Structure/2.07_Mass_Defect_-_The_Source_of_Nuclear_Energy