Is Pearson coefficient a good indicator of dependency between variables?
As you show, Pearson coefficient is clearly not a good measure of how variables depend on each other. A better measure is distance correlation. A nice property of distance correlation is that distance correlation 0 implies independence. An easy example that can happen in real life is that a variable is the square of another variable, as you show in your first picture. In this case, the Pearson correlation will be 0, but the distance correlation will be around 0.5. I think it is a mistake for Kagglers to choose variables in terms of Pearson correlations.