This question does not really make sense, since it mixes up different topics. What kind of correlation? Bivariat, linear or something like distance correlation, which is able to detect all kinds of relationships?

Hypotheses have nothing to do with analyses of relationships. You can do parameter estimation without hypothesis testing. So, what is the point of this discussion?

The purpose of 'fiding a relation' like in an analysis of variace is to be able to test whether the relation between two variables is a causal ralation and not a simple correlation. For example in applied medical research one wants to test whethter administering a drug has a possitive effect on the harling process. In such research there are usually 2 conditions, one with the relevant drug and one with a 'placebo' drug.

@ Statistical hypothesis testing ,

Finding of the relationship will provide a mathematical/statistical equation which is essentially important and useful in estimation, projection, forecasting etc. on the dependent variable(s).