As I said previously, it might not be as simple as an unweighted game of spin the bottle, but yes, that's pretty much the point.
Well, as georgie just pointed out, a pulsar is still a neutral star, and it isn't emitting nothing outside of the beam angles. We could very well detect it even if the beam doesn't hit us--the issue is, we would have trouble differentiating it as a pulsar and not a regular neutron star.That might be all the dark matter.