Welcome to Cross Validated!
I'm sure someone could give a more canonical answer, but here's the conceptual gist of it.
Think of it this way: there is only one null hypothesis, right? The hypothesis that there is no difference between your two samples/populations.
However, how could you define your alternative hypothesis? There could be infinite alternative hypotheses. If you have a statistically significant difference between your control and experimental group, that could be due to:
- Demographic (age, sex, ethnicity etc.) differences between your groups
- Difference in socio-economic status of your two groups
- Difference in diet between your two groups
- Difference in unknown underlying genetic factors
- Difference in epigenetic factors between the two groups, due to
varied prenatal/childhood experiences
- Difference everyday behavior (so the placebo didn't work, or worked
- The drug actually worked
I am not a researcher, and I don't know your experiment, so these are just guesses. But I hope you can see the point - there is only one null hypothesis (nothing is different between the groups) vs. an infinite number of possible explanations for an observed difference between the two groups. You assume that it's your intervention/drug that is the difference, but it could be anything! This is the conceptual reason why you don't test an alternative hypothesis - because which one would you test? What would be your assumed effect size? Too many (literally infinite) possibilities and assumptions. In vast majority of cases, the only thing you can actually test is "Is there some difference between these two groups?", and you set up the experiment as best you can so that if there is a difference, you can attribute it to the intervention (because you controlled for demographics, diet, behavioral factors etc.)
I hope this helped clarify your understanding!