Linearity is privileged mostly because it is the simplest type of relationship. This may seem arbitrary, but:

• It generalizes well. For example, if y is a polynomial function of x then y can be viewed as a linear function of the powers of x;
• Simple relationships have fewer free parameters and so should be favoured when selecting between models of comparable explanatory power;
• Simple relationships and linearity in particular have very nice mathematical properties that allow much deeper analysis than one might expect from their simplicity;
• A huge range of relationships are locally linear, in the sense of differentiability (which is a linear concept).

The first and last points in particular are used very widely in practice to great effect.

A researcher seeing the top-right chart is absolutely going to look for a suitable family of functions (such as polynomials where the relationship is a linear function of coefficients) and then use something like a least-squares method (which is based on linear error models) to find the best parameters and check how much variance (also based on linear foundations) remains unexplained by the proposed relationship.

I think you're basically right: Correlation is just one way of measuring dependence between variables. Being correlated is a sufficient but not necessary condition for dependence. We talk about correlation so much because:

1. We don't have a particularly convenient general scalar measure of how related two variables are. You might think about using something like mutual information, but for that you need the densities not datasets.
2. We're still living in the shadows of the times when computers weren't so big. We got used to doing all sorts of stuff based on linearity decades ago because we didn't have any other options, and they became "conventional" even when we might have better options now.

Spearman (rank) correlation is often a good alternative for nonlinear relationships.

Yes, in general the state of the art is more advanced than looking at correlations.

You just need to learn when using correlations makes sense. Don't assume that everyone is using correlations blindly; Statistics PhDs most likely decide whether to use them or not based on context and know the limited ways in which what the say applies.

Correlations make total sense when the distribution of the variables is close to multivariate Normal. The covariance matrix, which can be written as a combination of variances + correlation matrix, completely determines the shape of a multivariate Normal.

If the variables are not Normal, you can try to transform them to make them more Normal, using both univariate and multivariate transformations. This is a very common Statistics tool. Basic example: Quantile normalization.