Minimum sample size = 30 and the 10% rule for independence of trials

[Agent Smith]

Desmos graph

In the graph above, $\displaystyle y$ = Sample size and $\displaystyle p$ = population size.

1. The 10% rule for ensuring the independence of trials
$\displaystyle y \leq 0.1p$. Required for accuracy of the standard deviation (?)

2. The minimum sample size = $\displaystyle 30$ so that we can have a normal sampling distribution
$\displaystyle y \geq 30$. Required to ensure the sampling distribution is a normal distribution)

So ...

$\displaystyle 0.1p \geq 30 \implies p \geq 300$

To satisfy both conditions (which as far as I can tell is required in sampling distributions), the minimum size of the population $\displaystyle p \geq 300$.

Am I supposed to interpret this as statistical questions are meaningless for populations < 300? That is to say to do statistics we need a population of at least 300?