So for me the definition the independence of two random variables $X,Y$ is intuitivly very clear.

But what I have never seen motivated is why the heck one would be interested in the covariance $$\operatorname{Cov}(X,Y):=\mathbb{E}\left((X-\mathbb{E}(X))(Y-\mathbb{E}(Y))\right).$$ Since independent variables are also uncorrelated, covariance seems to be some weak form of independence.

Of course there are various nice results for uncorrelated random variables, for example $\mathbb{E}(XY)=\mathbb{E}(X)\mathbb{E}(Y)$ or various forms of convergence results like the law of large numbers or the central limit theorem. But what is the essence of covariance, what intuitive notion does it capture? Every example of uncorrelated yet dependent random variables seem very contrived to me. What would a "real-life-example" of two such variables look like?