One thing I don't understand about what this page seems to suggest: shouldn't there be a bright ring of starlight at some non-zero angle away from dead ahead?

Given a finite collection of objects out to a certain radius (stars), relativistic length contraction will compress it along the direction of travel, so an observer looking out from the centre should see the density increase to a maximum when perpendicular to the contracted direction (in a way that's sort of the opposite of synchrotron radiation ending up tightly directed forward and backward). I guess the aberration described in your link will bend this fore-wards from the perpendicular, but it seems like it should still be visible.