The possible discovery of a fast pulsar in the binary Cygnus X-3 is usually believed to be in keeping with the earlier conjecture that the gamma-ray luminosity is fed by the rotational energy of a young and fast pulsar. However, several pieces of circumstantial evidence are presented here to draw a contrary conclusion. We emphasize that if the X-ray flux indeed results from accretion, then the possibility that the pulsar generates a strong wind comprising among other things a superstrong proton beam which is usually invoked to explain the ultrahigh energy (UHE) gamma-ray observations from Cygnus X-3, seems to be an incongruity. It is indicated that the idea that the pulsar in Cygnus X-3 is in fact an old recycled one fits neatly in the general framework of evolution of X-ray binaries toward binary millisecond radio pulsars. Simultaneously we point out that no canonical evolutionary scenario is available at present which can explain a near-Eddington mass transfer rate in an X-ray binary with a period of few hours. It is suggested that the mass loss from a low-mass helium subgiant (instead of an H subgiant which would be too large) evolving on a nuclear time scale can be a viable way of attaining a prolonged mass transfer at a rate greater-than-or-similar-to 10(-8) M* yr-1 for short-period systems. Accordingly, we present a canonical picture of an evolutionary scenario for Cygnus X-3 in which the companion is a stripped He core of a massive star, and in which case the mass of the companion is M(c) approximately 1 M*. Our model allows for the possibility that the neutron star could be old enough to have a surface magnetic field B(s) less-than-or-similar-to 10(10) G. However, if the observed unusually high value of P/P = 2.2 x 10(-6) yr-1 is a real effect and due to mass loss from the binary, then the above canonical picture fails, and a much more unconventional scenario in which the companion is much lighter and undergoing a more rapid mass transfer caused by some bootstrap mechanism is called for. In this case too, the neutron star must be a very old one.