Can regression replace correlation? Could one explain the value of correlation in statistics?
If regression is a so advanced technique, why would we even need correlation? Could we also use theoretically use regression for measuring a degree of relationship between two variables? What is so special about correlation? Just a convenient, standardized statistic for showing degree of relationship between two variables?

You’re right: correlation drops out of regression.
If we have just two variables, $X$ and $Y$, we get $cor(X,Y)$ form the square root of the regression $R^2$ and the sign of the slope coefficient.
However, correlation gives an intuition about what a strength of relationship means (particularly since the word has entered into common English), and it skips regression issues that beginning students are unlikely to follow, such as matrix algebra and minimizing the sum of squares (calculus). We don’t need to go that deep to teach correlation to middle school students, who will follow correlation but perhaps not the math of OLS.
Note, however, that an awful lot drops out of regression. Even a t-test can be formulated as a regression on a binary variable, so correlation is not unique in this regard.
(And Spearman correlation drops out of ordinal regression, and the $\chi^2$ test drops out of logistic regression. My professor wasn’t kidding when we said that basically everything in statistics is regression!)
