How random is the graduate admission process in the United States?
There is without a doubt some randomness in the process, but I find your question impossible to answer. In the question, you implicitly assume that the "quality" of an application is a well-defined concept that could be used to replicate an experiment.

But this isn't the case. The only way you could do this is by submitting N "equally good" applications to the same graduate programs in the same year, and expect the same outcome for all. But this is clearly not a workable concept. Rather, different graduate programs may look for different candidates (for example because they have different areas of strength), and possibly may want to balance areas in successive years. In other words, the "quality" of an application is a concept that can only be measured against the current needs of individual graduate programs. You can't repeat the experiment at a different place or different time and expect the same outcome.