This is a bad explanation of p values. Of the start:
> If you’re testing whether mean=0 and actually the truth is that mean=0.000000001, and if you have a large enough sample size, then YOU WILL GET A TINY P-VALUE.
Think about this for a second. Does it make sense that the results of my statistical test would change if I measured in terragrams vs. micrograms? No, it doesn't, because it's not the numerical difference between null hypothesis and sample mean, it's the difference relative to standard deviation.
If the effect size is 0.000000001 and the standard deviation is 0.0000000000000001 then this is an enormous, important effect. If the standard deviation is 0.1, then this is a really small and unimportant effect. It can still be observed correctly with a sufficient sample size though because you've stated it's a real effect to begin with.
You will absolutely not get a tiny p value just because your sample size is large. That is wrong. You will be able to detect very small effects, but if there is no difference, you are more likely to draw the correct conclusion with 1,000,000 samples than with 100 samples.
The point the author here is trying to convey but is not getting clearly, is that a large sample size can detect an effect that is practically useless and call it significant. p-values tell you if it seems probable, given your data, if there is a difference between the two datasets. They make no claims about whether the difference is material.
Example python code to do prove p values are not likelier to be smaller just due to large sample sizes, given no actual difference:
import random
from scipy.stats import ttest_ind
num_tests = 10000
res = []
for test in range(num_tests):
n = 100
a = [random.random() for _ in range(n)]
b = [random.random() for _ in range(n)]
res = res + [ttest_ind(a,b)[1]