The violated assumption is simply i.i.d. of the so-called "random variable", in this case the individual coin flips would need to be i.i.d. for the CLT to hold.

The violated assumption is simply i.i.d. of the so-called "random variable", in this case the individual coin flips would need to be i.i.d. for the CLT to hold. Obviously that's not the case at all by construction of the experiment. So really, it would be extraordinary if the plot showed anything other than it does. While the explanation of why it doesn't apply was handwavy ("can't really be written as a sum in the same way as before"), it is certainly correct and the graph helps give the intuition.