It is my opinion only, but I feel that Stanford does not have as strong of an astronomy program as other California schools. For example, when I think of "top California astronomy programs", UC Berkeley, UC Santa Cruz, and Caltech come to mind first. I get the sense that Stanford's top ranking in Physics/Astro programs is more due to their Physics program than their Astronomy program.
However, that was all opinion/speculation. When I try to look up Stanford's astronomy department, it seems like they don't really have one (http://web.stanford.edu/dept/astro/
). This webpage makes it seem like there is no actual astro department, but instead, it is just a research area that Physics, Applied Physics and Electrical Engineering departments pursue.
I feel that this signals to me that Astronomy is not a serious priority for Stanford. Although many schools do not have a separate Astro department, most astronomy programs are at least housed in a "Physics and Astronomy" department. However, at Stanford it seems like Astronomy is "just" a subfield of Physics. And from their Physics page, it does seem like Stanford does not cover the whole gamut of astronomy fields, but instead seems to mostly focus on Astrophysics. When I check the courses available to Physics PhD students, I get the same sense.
Finally, as far as I know, Stanford does not operate or have time on 10m class telescopes (e.g. Keck). I think they do operate some instruments/telescopes that are useful for particle astrophysics and/or cosmology research but not necessarily useful for the rest of astronomy. Thus, this might explain the small number of astronomer profiles you see here applying to Stanford.
In summary--I think Stanford gets less mentions here because unlike other California astro programs, it is more specialized and only covers a small number of research interests, so naturally, the number of people interested in Stanford is smaller! Personally, I did not apply to Stanford because Stanford's specialities and resources do not overlap with what I wanted! A good example, I think, of how it's important to consider research fit as well as department ranking!