Are we misuse the t-test in randomized experiment? I am reading the Rubin's Causal Inference book and I found his "potential outcome" framework really fits my tastes. Then I rethink about the t-test we often used in randomized experiment setting.
Suppose we have 20 people and we want to know the difference between treatment $A$ and $B$ on heart rate, for example. 
Then according to Rubin's potential outcome framework, these 20 people will have a distribution of heart rate if everyone got treatment $A$, denote this distribution as $F_A$. Similarly, if all of these 20 people get treatment $B$, then there will be another heart rate distribution, denoted by $F_B$. Note that we, as experimenter only observe one outcome for each people.
Now, for each people, we flip a coin to determine whether this people get treatment $A$ or $B$. Say in the end, we get 11 people take treatment $A$ and 9 people take treatment $B$. Then we measure their heart rate, then we compute the average heart rate within each group, then we conduct t-test done. 
Note that these 11 people got treatment A can be seen as 11 draw from the distribution $F_A$. However, these 11 draw are NOT independent. The reason is because after you draw one guy, you cannot draw that guy for the second time. Hence, this is actually 11 draw WITH REPLACEMENT from the population $F_A$. One can show that these 11 people, whose heart rate is 11 identical but not independent draw from the population $F_A$. Similarly argument applies for the other 9 people who take treatment $B$.
Then my question is: for using t-test, I remember it requires the population distribution is normal and the draw is i.i.d. It seems that in this example (and I believe it represents many practical experiments) I made, neither of these assumptions holds. So are we misuse the t-test? Or did I miss something?
 A: As far as normality is concerned, this can be helped by increasing the sample size to 60.
To your main question:
If you see the 20 people as the population to draw the two samples from, then your reasoning is correct. You should not see your 20 people as the population to draw from however. Doing this would mean that the sole purpose of the research is to potentially help those 20 people with a future treatment. Surely, you want to sell this treatment to more people than just the 20 test subjects!
The population to draw from is everybody who might be helped with this treatment. When looking at this much larger population (assuming we are not talking about some extremely rare disease), it makes not much of a difference to sample with or without replacement. For example, with a population of 100000, does it really matter that your second sample group  is drawn from a slightly smaller population of 99990 after the first ten subjects received treatment A? Does subtracting 10 subjects from 100000 really change the population distribution? This can be neglected in favor of a between subjects design where no person is measured twice.
The random sampling is effectively done in two steps:


*

*Sample a group from the general population

*Divide group into treatment A and B


I would be more worried about potential non-random sampling in step 1 than in step 2. (With low response rates, it could be that only people with a specific prior interest sign up for the experiment. Are you sampling from all geographic areas where you want to sell the treatment or only locally for convenience? ...)
If steps 1 and 2 would be done right away for each candidate as soon as he signs up, would you still be worried about $F_A$ and $F_B$? There would not be an $F_A$ and $F_B$ to begin with. Are random number generators sufficiently random that it doesn't make a difference to flip coins one by one when the candidates arrive or all at once when all candidates have been signed up? We should hope so, much depends on it.
