Random selection would also likely have a moderating influence on the courts of appeal, because an appeals panel, intent on adopting an extreme position, could no longer expect review by a politically, ideologically, or philosophically sympathetic Supreme Court.
Isn't the opposite just as likely to be true? A random Court with little institutional memory and no control over the decisions in the cases it does decide to hear is likely to only hear a very few cases. An ideological circuit could decide to just play the odds that its number won't come up. Certainly, they can do that now too, what with the limited amount of cases the Court hears. But this would exacerbate the problem that much more, in my opinion.
Furthermore, the way it is now, the interplay of the dominant, but distant Supreme Court, and the workhorse circuit courts inherently moderates judicial decisions. Think about it, how many earth-shattering decisions have come down from the Supreme Court even in the last century? Brown, Roe, maybe one or two others. And even those have had a far more limited effect than normally ascribed. Critical Legal Theorists have noted that the combination of administrative delay, lower court circumscription, and other factors have severely limited the effect of even the most landmark cases. Thus, as many students are in segregated schools today as there were in Brown, and for many women, getting a safe abortion is not much easier than it was in the pre-Roe era. And while this has it's problems, it is also somewhat predictable, and thus respondable. Kaiser's proposal would throw in yet another level of randomness to solve a problem I don't think exists. Frankly, the myth of an evil, overarching, activist court is just that, a myth. Rarely, if ever, has a court decision made an impact on any of our lives, and it's rulings have made narrow, if any, indentation of the more important political issues discussed around the nation. The whole "attack the court" mentality is a function of conservative victimization--the deluded mindset that their values are under assault even as they control all branches of the federal government and the majority of local governments as well. Weakening the Court would do little to address any substantive concerns (such as they are), it is an ego shot designed to put more power in the institutions that conservatives currently control. When a liberal program comes out through democratic institutions that conservatives wish to see buried, they'll be the loudest advocates of anyone for Supreme Court intervention (see, e.g., Kelo).