Under General Relativity, Lorentzian wormholes (the kind that are traversable) require exotic matter (a kind of unobtainium which is not known to exist).
This is not true. A maximally extended Kerr black hole solution for instance has a traversable wormhole to a (different) external universe and doesn't require exotic matter. However you do have to traverse a region with a visible singularity and with visible regions where time travel is possible (where closed time like curves exist). It is unreasonable because it is an eternal solution so the black hole never formed from infalling matter.
On the other hand, we know black holes exists and these form from the collapse of large stars.
External observers, by definition, never see them form.
One of the differences that we usually associate with wormholes is that they do not have an event horizon, and are disallowed by topological censorship theorems.
Most topological censorship is based on changes in topology. And there are traversable wormholes with event horizons, but that is usually interpreted meaning they go to different universes and the alternative seems to be closed time like curves that go through the wormhole. And the latter happens anyway, just if they are different universes the closed curves loop around only in the inside regions and never in the outside regions.
But what about black wormholes? is this concept a meaningful one? what I'm thinking is a Lorentzian wormhole that is unidirectional, basically you fall in an event horizon, but instead of finding a singularity, you enter a throat and exit on the other side.
This seems lime a novel (possibly nonexistent) solution if you are trying to avoid using exotic matter.
From the exit mouth, you can 'see' the other side, you can even traverse the wormhole back to the entrance side, but you are unable to send anything to null infinity because of the event horizon.
This is very confusing. If you can cross a surface going in both directions it doesn't sound like an event horizon.
Would such exit side be essentially the same thing as a white hole?
White holes can have there own type of event horizon, ones where you can cross from the inside to the outside but not vice versa. This is the usual type of horizon to cross into an external universe, such as in the maximal Kerr solution.