We first advance a mathematical novelty that the three geometrically and topologically distinct objects mentioned in the title can be exactly obtained from the Jordan frame vacuum Brans I solution by a combination of coordinate transformations, trigonometric identities and complex Wick rotation. Next, we study their respective accretion properties using the Page–Thorne model which studies accretion properties exclusively for rge r_{text {ms}} (the minimally stable radius of particle orbits), while the radii of singularity/throat/horizon r<r_{text {ms}}. Also, its Page–Thorne efficiency epsilon is found to increase with decreasing r_{text {ms}} and also yields epsilon =0.0572 for Schwarzschild black hole (SBH). But in the singular limit rrightarrow r_{s} (radius of singularity), we have epsilon rightarrow 1 giving rise to 100 % efficiency in agreement with the efficiency of the naked singularity constructed in [10]. We show that the differential accretion luminosity frac{d{mathcal {L}}_{infty }}{dln {r}} of Buchdahl naked singularity (BNS) is always substantially larger than that of SBH, while Eddington luminosity at infinity L_{text {Edd}}^{infty } for BNS could be arbitrarily large at rrightarrow r_{s} due to the scalar field phi that is defined in (r_{s}, infty ). It is concluded that BNS accretion profiles can still be higher than those of regular objects in the universe.