I have read the answer here. Here the distinction is that
If $n\to\infty$ and $p\to0$ while $np$ approaches some positive number $\lambda,$ then the binomial distribution approaches a Poisson distribution with expected value $\lambda.$
If $n\to\infty$ as $p$ stays fixed, and $X\sim\operatorname{Binomial}(n,p)$ then the distribution of $(X-np)/\sqrt{np(1-p)}$ approaches the standard normal distribution, i.e. the normal distribution with expected value $0$ and standard deviation $1.$
I am finding it hard to wrap my head around this. In the derivation of the central limit theorem nowhere is $p$ taken into consideration. So even if $p$ is very small, according to CLT the standardized Binomial should limit to a standard normal. And the two limiting behaviors are both for $n \to \infty$ Please help me understand this concept a bit more. How can CLT not be valid when $p$ is really small?