Why does the autofocus capability depend on the aperture rather than the amount of light available?
Because the way phase detection autofocus works is dependent upon the difference in the light rays from the same area in the field of view as they strike opposite sides of the lens. The wider the effective aperture of the lens (more properly called the entrance pupil), the wider apart the light rays that are compared can be. The wider the difference between the two points, the better AF systems can perform in terms of accuracy and and speed. More light helps too because it increases contrast between lighter and darker edges, but only if that brighter light is reaching the lines on the PDAF sensor array.
Each AF point uses a pair of lines of light sensitive pixel wells similar to those found on the camera's sensor. Each pixel well is (generally) larger than the main image sensor's pixels and they are not filtered for color differentiation like a Bayer masked sensor is.
To take advantage of the wider aperture, the pairs of line sensors for a particular focus point in the AF array, or more precisely, the microlenses at the entrance to the PDAF sensor array that aim light at the lines on the surface of the PDAF sensor, must be further apart from one another. But that makes those lines not very useful when a lens with a narrower aperture is attached to the lens, because then no light is reaching those points. Some camera manufacturers hedge their bet a little. Some of the focus points are more sensitive/accurate but only function well with a large aperture lens. Other focus points are tuned to be able to use the light from lenses with narrower apertures. But those points can't take advantage of the wider light rays provided by a wide aperture lens.
This is because the two lines on the focus array for each focus point are in a fixed position. If they are close enough to each other to be able to use light that gets through each side of the lens with a narrow f/8 aperture, they are not far enough apart from each other to sense the light that gets through the edge of the lens with a wide f/2.8 or wider aperture. Even when a faster lens is on the camera they are only using light falling on parts of each side of the lens that are close enough to the center for that light to make it through the narrower aperture.
All PDAF focus points use a pair of lines. Cross type focus points use two pairs of lines: one pair for the vertical and a separate pair for the horizontal. "Diagonal cross type points" add an additional two pairs of lines oriented at 45° angles to the vertical and horizontal lines.
The distance these lines are apart on the PDAF sensor array (combined with the way the microlenses on the entrance of the PDAF array aim the light falling on various parts of the front of the camera's main lens) determine how far from the center of the lens' optical axis is being sampled by that set of lines on the PDAF sensor. If the two lines for an AF point are sampling light from two opposite points on the front of the lens that are closer to the center of the lens' optical axis they will work at narrower apertures, but they will be less sensitive. If the two lines for an AF point are sampling light from two opposite points on the front of the lens that are further from the lens' optical axis they will be more sensitive, but they will only work if the lens has an entrance pupil wide enough that there is actually light making it to those lines on the PDAF sensor array.
Camera makers, particularly Canon which is indirectly referenced in your question, hedge their bets a little. This is especially the case with PDAF systems that have a very large number of AF points. Some of the points are tuned to sample light that is passed by narrower lenses, such as f/5.6 or f/8. Other points are tuned to look further to the edge on opposite sides of the lens. These points only receive light from the lens if the entrance pupil is wide enough to allow light to reach them.
In the past, Canon tended to make each set of lines in their cross type points sensitive at different apertures. The horizontally sensitive set of lines may be sensitive to lenses with a maximum aperture of f/5.6 or wider while the vertically sensitive set of lines for the same AF point may only work with lenses having an f/4 or wider aperture. A little more recently the cutoff for each pair in the set may be f/8 and f/5.6, respectively. Some of the more recent high end models have crosspoints where both sets of lines for a single cross type AF point are capable with lenses or lens/extender/teleconverter combinations as narrow as f/8. Improvements in the technology of AF systems have allowed the smaller differences detected by sampling narrower baselines to result in performance that could previously only be obtained from AF points using a wider baseline.
Those same improvements can also be used to make f/2.8 AF points better! Canon also tended (and still do) to make the diagonal cross-type points use a wider baseline that requires a wider aperture lens, usually f/2.8 or wider. This gives those points extremely accurate sensitivity where it is needed the most: when used with wide aperture lenses that give an extremely shallow depth of field and very little margin for AF error.
Since "cross type" AF points are really two sets of lines sensitive at 90° angle to each other, there's nothing that says you can't make a cross type AF point, or even a diagonal cross type AF point, that will work with lenses having an aperture narrower than f/2.8, f/4, f/5.6, etc. It's just that camera makers, particularly the one referenced in the quote contained in the question above, have chosen to make their diagonal AF cross points their most sensitive ones.
Incidentally, the same physics that favors AF with lenses with wider apertures also favors DSLR cameras with larger sensors. Because the mirror is larger, particularly because it is wider, and allows the semi-translucent portion that allows light through to the secondary mirror to be reflected down into the PDAF array to also be wider in a full frame camera than in an APS-C camera, the baseline used for the most sensitive focus points can also be wider.
For a little deeper answer on how cross type points work and a visualization of how f/2.8 points require lines that are further apart, see this answer.
For a slightly different take and a couple of more thorough looks at how the maximum aperture can affect AF performance (as compared to How to enable Canon AF with teleconverter?), please see the two upvoted answers to: Will the Canon 5D MK II with 100-400 1:4.5-5.6 work properly with Kenko 1.4 converter?