The derivation of the recently proposed one-component relativistic Hamiltonian, and the resulting relativistic scheme by eliminating small components (RESC) method of Nakajima and Hirao, are analyzed in terms of the Foldy-Wouthuysen transformation of the Dirac Hamiltonian. This approach reveals the meaning of different approximations used in the derivation of the RESC Hamiltonian and its close relation to approximate relativistic Hamiltonians resulting from the free-particle Foldy-Wouthuysen transformation. Moreover, the present derivation combined with what is called the classical approximation in Nakajima and Hirao's approach shows that there is a whole family of the RESC-type Hamiltonians. Some of them, including the original RESC Hamiltonian, are analyzed numerically. It is documented that neither of the RESC-type Hamiltonians offers variational stability. As a consequence the RESC methods may suffer from the variational collapse for heavier systems. On the other hand the energy differences (e.g., ionization potentials) computed within the RESC approach turn out to be close to the values obtained in the Douglas-Kroll scheme.