With conventional Phase Detection AF systems found in SLRs the width of the entrance pupil of the lens is one of the factors that determines the baseline: how far apart the rays from each side of the lens are from each other. The wider the aperture, the wider the baseline. The wider the baseline the more difference there will be between out of focus light from each side of the lens. This allows the camera to focus more accurately and with less light if the focus array is designed to take advantage of the wide baseline provided by a large aperture lens. But if the PDAF focus array is only designed to take advantage of f/2.8 lenses, putting an f/1 lens on the camera would make no difference in that regard because the rays closer to each edge of the lens than those admitted by an f/2.8 lens would not be falling on the PDAF sensor.
As a result specifications for focusing performance in low light have traditionally included the aperture required to allow the camera to perform at a certain EV level. But with conventional mirrorless cameras there is no comparison between the rays coming from one side of the lens to the rays coming from the other. Instead, contrast detection AF is used. The maximum aperture of the lens is still important, however, because the amount of light admitted by an f/2.8 lens is more than the amount of light admitted by, for example, and f/4 lens. Twice as much light, in fact. So if your camera can focus in conditions requiring EV 0 to expose correctly with an f/2.8 lens, it would need conditions of EV 1 to focus with an f/4 lens, EV 2 to focus with an f/5.6 lens, and so on.
Newer mirrorless camera models often include hybrid CDAF. This is the case for the Sony A600 reviewed at the link in your question. Some of the sensor's pixels are masked in such a way that they only collect light rays from one side of the lens or the other. The data from these special pixels is used to do PDAF. How wide the baseline is will affect the performance of such systems to one degree or another.