What is the most important satire: the patients who look for particular answers or for authority to hand down diagnoses, the officials responsible for law and governing, or the psychiatrist himself?

"The Psychiatrist" could be interpreted as a satire on positivism. This is a philosophy that predominated in the West in the late 19th and early 20th centuries which held that all human problems would one day be solved by science. At the time Machado wrote the story, positivism had spread to his native Brazil, becoming almost an article of faith among the educated middle classes. Then, as now, psychiatrists believed themselves to be practicing science and saw their work with patients as involving the application of insights gleaned from scientific theory.
The learned psychiatrist Bacamarte—which in Portuguese means "good for nothing"—acts as if his scientific training has given him a status akin to a priest in pre-Enlightenment times. Bacamarte's high-handed behavior and insistence on his absolute authority as a professional reveals a surprising kinship between the men of science and their clerical predecessors. Positivism is supposed to be based on a fidelity to scientific rationality, yet in the figure of Bacamarte, it displays the kind of rigid dogmatism one normally associates with a certain kind of religion.
The psychiatrist of the title arrogates to himself the right to determine—in the broadest possible interpretation—who is and who isn't insane. The enormous power that this gives him makes him almost infallible in his own professional field. Again, the parallel with clerical power isn't difficult to detect. It's as if, in positivism, humankind has simply replaced one dogmatic religion with another.