An infinite-U term is introduced into the Holstein - Primakoff-transformed magnon Hamiltonian of quantum Heisenberg magnetic models of spin s. This term removes the unphysical spin-wave states at every site, and automatically truncates the expansion in powers of the magnon occupation operator. The resultant strong-coupling magnon Hamiltonians are accurately equivalent to the original spin Hamiltonians. The on-site U-levels and their implications are studied. Within a simple decoupling approximation for our strong-coupling magnon models, we can easily reproduce the results for the (sublattice) magnetizations obtained previously for the original spin model. But our bosonic Hamiltonians without any unphysical states allow substantially improved values to be obtained for the spectral weight in the ground state, and for ground-state energies lower than those obtained within previous approximations.