  We study the evolution of newborn neutron stars in high-mass X-ray binaries
interacting with a wind-fed super-Eddington disk. The inner disk is regularized
to a radiation-dominated quasi-spherical configuration for which we calculate
the inner radius of the disk, the total luminosity of the system and the torque
acting on the neutron star accordingly, following the evolution of the system
through the ejector and early propeller stages. We find that the systems with
$B \gtrsim 10^{13}$ G pass through a short ($\sim 20\,{\rm yr}$) ejector stage
appearing as supernova impostors followed by a propeller stage lasting $\sim
10^3\,{\rm yr}$. In the super-critical propeller stage the system is still
bright ($L\sim 10^{40}\,{\rm erg\, s^{-1}}$) due to the spindown power and
therefore appears as an ultra-luminous X-ray source (ULX). The system evolves
into pulsating ULX (PULX) when the neutron star spins down to a period ($P\sim
1$ s) allowing for accretion onto its surface to commence. Systems with lower
magnetic fields, $B \sim 10^{11}$ G, pass through a long ($10^5\,{\rm yr}$)
super-critical propeller stage with luminosities similar to those of the
ultra-luminous super-soft sources (ULS), $L \lesssim 10^{40}\,{\rm erg\,
s^{-1}}$. The equilibrium periods of these systems in the accretion stage is
about $10\,{\rm ms}$, which is much smaller than the typical period range of
PULX observed to date. Such systems could have a larger population, but their
pulsations would be elusive due to the smaller size of the magnetosphere. Our
results suggest that the ULS and some nonpulsating ULX are rapidly spinning and
highly magnetized young neutron stars at the super-critical propeller stage.
