Is there a simple metric (Lorentzian manifold) known which exhibits the formation of a black hole while not having any white hole counterpart and which moreover satisfies the strong and dominant energy conditions at every point? I know of the Oppenheimer-Snyder model (although there is rumour that this model also includes a white hole, which sounds plausible to me), but I'm looking for something more straightforward where moreover geodesic completeness (excluding geodesics which end on the singularity of course) can be checked at a glance.
If you are looking for "Simple Analytic Models of Gravitational Collapse" then you can look at the paper of the same name by R. J. Adler, J. D. Bjorken, P. Chen, and J. S. Liu in the American Journal of Physics: 73 (2005) 1148-1159 https://dx.doi.org/10.1119/1.2117187 (preprint at arXiv:gr-qc/0502040).
The easiest possible example is if you imagine a shell of matter moving inwards at the speed of light with Minkowski spacetime inside.
So empty Minkowski spacetime on the inside. A fixed, constant mass Schwarzschild spacetime on the outside. And a shell of source at the interface between the two. And as you go back in time the shell is simply farther and farther out. So it can be far out, but was never a white hole.
Now someone can always place an event horizon on the conformal infinite past. But that can happen with empty Minkowski space, so it can't mean very much, plus you can put different topologies and dimensions on the conformal infinite past or even make it be empty, so how much do you really want to worry about it.
As for the white holes in some Oppenheimer-Snyder models: they are for initial conditions with insufficient kinetic energy. Roughly, you want to give your material enough initial kinetic energy that it falls in from spatial infinity with some finite kinetic energy in the infinite past. Basically if you push a moment of time symmetry way back all the way into the infinite past then you can ignore a white hole the same way Minkowski spacetime does.
As for whether this can be done generically via a small perturbation, that can depend on what you consider to be small. And it is very similar to advanced solutions from electromagnetism.
A large shell of dust at rest has only slightly less energy than a shell of dust that fell from rest at infinity. But a small ball of dust at rest can have a lot less energy than the same dust with enough energy to have fallen from spatial infinity in the infinite past. But to an outside observer at any moment they just see a Schwarzschild solution with parameter $M$ and they would be equally happy with one where the dust in the ball has less rest energy in each section and more kinetic energy. And I'd say that is more physical exactly because it doesn't come from a white hole.
But really, dust is not very realistic anyway. But is rather hard to be realistic as well as simple. A real star is going to emit radiation and hence get smaller over time unless energy is flowing in. So you could set up a realistic star that has energy flowing in exactly equal to what is flowing out. Or try to model a ball of gas so cool it doesn't have fusion with a simple thermal infall equal to its thermal radiation. But that isn't really formation. If you made it an agonistic model instead of a fluid model you could have gas molecule from the too cool for fusion ball sometimes get escape velocity and have a flux of particles from infinity.
But then that's the same issue. To counter a flux of particles with escape velocity, you need to supply replacement particles with enough energy to have fallen from infinity with some finite kinetic energy. Exactly what would have worked for dust. And this is still just maintenance, not formation.
But that seems to be the real issue. There are stars that were essentially an infinite time in forming. And stars that were always around. And stars that formed out of white holes.