Mosley, I'm going to give you the benefit of a doubt. Perhaps you just misworded your statement. Search engines don't "produce" anything. They can bring you results, but they do not produce content.
Maybe he used "produce" in a different sense of the word. "To produce" can mean to make available or to present for view, as when "producing" evidence or a witness. In that context, you don't create evidence or witnesses, you find them and let other people access them. If that's what he meant, "bring you results" isn't too far off.
Because patent lawyers are not in the business of rendering anything unpatentable
Just to be clear, Mayo was arguing here that certain things should be unpatentable. And one of the lawyers whose name is on Mayo's brief is a registered Patent Attorney employed by Mayo. So for the record, at least one patent lawyer was trying to "render [some]thing unpatentable" here.
More generally, though, patent counsel representing the defendants in infringement suits make their very livings trying to render things unpatentable.
The patent wasn't for "a specific way of testing for a disease," and Prometheus isn't the patentee. The method in question is [purportedly] therapeutic, not diagnostic; if you accept Prometheus's interpretation of it, it describes a way to safely administer a dangerous drug without killing your patient. And Prometheus is the exclusive licensee of the alleged invention, not the inventor.
Also, it should be noted that the AMA's brief doesn't oppose the patentability of diagnostic tests or medical devices or processes generally, as you suggest:
Amici medical associations recognize that health-care-related patents can enhance the provision of high-quality and cost-effective medical care. The financial incentive that patents offer supports the expensive and uncertain research required to identify, test, and gain approval for new pharmaceuticals, medical devices, diagnostic testing kits, and other products. In this respect, the patent system has served patients and the medical profession well, drawing investment into the development of important new treatments.
Patents on scientific observations underlying medical care, however, do not have these salutary effects. Such patents erode the quality of patient care by limiting use of the very knowledge on which physicians must rely to diagnose and treat their patients, threaten to stifle innovation and the development of personalized medicine, and raise ethical concerns for physicians.
(1) to perform or display it at a place open to the public or at any place where a substantial number of persons outside of a normal circle of a family and its social acquaintances is gathered; or
(2) to transmit or otherwise communicate a performance or display of the work to a place specified by clause (1) or to the public, by means of any device or process, whether the members of the public capable of receiving the performance or display receive it in the same place or in separate places and at the same time or at different times.
Granted this is frustratingly circular, what with the way it defines "public" by reference to the "public". But as the court points out, Congress's definition specifically does include a closed off back room accessible to a succession of private parties, even if only a single party at a time has access. So contrary to what you wrote (which does make perfect English sense), the law defines a public place by reference to people as members of the public.
Mind you, I'm not defending the language of the statute, just pointing out that the court's decision is much more reasonable in the context of interpreting that statute—which was intentionally made broad enough to include hotel rooms as public places.