Not quite correct, in your analogy the demonstration of one black swan does prove that at least one black swan exists. This is because everyone can agree that the swan is black.

In a clinical trial, the results have to be examined statistically. Why, you might ask? The answer is that many other factors may influence the outcome after treatment, plus the general variablility of biological systems. The standard in clinical trials is that if the results are more likely than a probability of 5%, they are considered significant. A problem with this test of significance is that if you tested pure water against pure water for a disease then 5% of the time pure water will be better than pure water at the level of 5% significance. This is a consequence of the mathematics of randomness.

When there is an accepted mechanism for action in any part of science and a single experiment (trial) shows that there is a significant probability that it shows the correct effect, it is incorporated into the corpus of knowledge. If the experiment significantly shows that the accepted mechanism is incorrect, it will be viewed with suspicion and will need to be repeated by different people to confirm the result. This is how science has advanced and achieved such stunning results.