A/B Testing (Binomial Distribution vs Random Distribution)
The question you are asking is somewhat confusing and I believe there's a few assumptions you are using which are incorrect.  Please bear with me and correct me if I'm misinterpreting.
You suggest there are different types of click thru rates, constant and random (normally distributed); This is not true.  Click thru rates are assumed to be randomly distributed in a sample.  I do not know what a constant click-thru rate is, but there is no such assumption in the T-test calculation.  Order of clicks do not matter, only the probability, which is equivalent to the actual click thru rate (which is unknown).  When you sample, through measurement, you will then calculate the estimated click thru rate.  We know that this process is probabilistic, and we can end up with estimates that are higher, or lower than the actual CTR, but it should be close.  It is the sample mean which is normally distributed as described by the Central Limit Theorem.  The standard deviation of the estimated CTR, also known as the standard error, is set by the estimated CTR.
The &quot;distribution&quot; of clicks in your sample is assumed to be random, but the order should not matter in the calculation, and should not affect the validity of the test results.
Running an AA test is a good way to confirm whether all the math, statistics, and logging are working out.  If you are finding statistically significant results more frequently than your significance threshold (alpha), something is wrong.  But I cannot see how the &quot;distribution of clicks assumed to be constant&quot; can possibly be the root cause.
Good luck and may your tests be properly powered.