A new analysis of the precise experimental astrophysical S factors for the direct-capture reaction ⁷Be(p, γ)⁸B [A.J. Junghans et al., Phys. Rev. C 68, 065803 (2003) and L.T. Baby et al., Phys. Rev. C 67, 065805 (2003)] is carried out on the basis of a modified two-body potential approach in which the direct astrophysical S factor, S ₁₇(E), is expressed in terms of the asymptotic normalization constants for p + ⁷Be → ⁸B and two additional conditions are involved to verify the peripheral character of the reaction under consideration. The Woods-Saxon potential form is used for the bound-(p + ⁷Be)-state wave function and for p ⁷Be-scattering wave function. New estimates are obtained for the “indirectly measured” values of the asymptotic normalization constants (the nuclear vertex constants) for the p + ⁷Be → ⁸B and S ₁₇(E) at E ≤ 115 keV, including E = 0. These values of S ₁₇(E) and asymptotic normalization constants have been used for obtaining the indirectly measured values of the s-wave average scattering length and the p-wave effective-range parameters for p ⁷Be scattering.