Saying that correlation is usually accidental is a stretch. Basing that claim on that (admittedly amusing) website is even more of a stretch: the time series on that website are all highly autoregressive with minuscule sample sizes, which makes spurious correlations extremely likely [1].

Speaking (very) roughly, the fact that they are autoregressive constrains how 'kinky' the shapes can be. Visually speaking, the time series will have a small number of inflection points and will appear interpolated between these points.

This greatly reduces the search space: you just need to line up a couple of kinks in a space of ten samples, or match two smooth and vaguely line-shaped objects, rather than find a convincing relationship for 100s of observations not tied together in time.

[1] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.611...

I see your blog post and raise you a youtube video.

minutephysics Correlation CAN Imply Causation! | Statistics Misconceptions

https://www.youtube.com/watch?v=HUti6vGctQM

I think if you measure random unrelated data sets against each other you're quite likely to find spurious correlations. If you design an experiment between potentially related variables with a potential hypothesis on how they are related and then observe data that matches the hypothesis, you probably have a reasonable chance the hypothesis is actually the explanation.