Intragroup indepence in two groups analysis

I am working in an experiment in which I want to analyze the impact of a treatment on two different groups of customers. Most of the method for analysis I have checked (for example t-test) have as a hypothesis the existence intragroup and crossgroup independence. I can assume the crossgroup independence because the two groups are randomly split, but I have some doubts about the meaning of the intragroup independence.

We can assume that there is no causal effect of the behaviour of a client on the behavour of some others. But it is very likely that similar clients will behave very similar (younger clients and older clients will behave very similar to people in the same group and very different to people on the other). Is this enough to make the test assumptions fail?

In order to check if the intragroup independence assumption is right, I am taking N random samples of the mean of K values for high N and K and I check if the distribution is normal using Kolmogorov–Smirnov test. According to the central limit theorem, if the samples are independent the sample average will follow a normal distribution. If the test fails, I know the samples are not independent. Is the reasoning correct?