First of all, decision Trees can easily use categorical features with entropy and gini index, but (Pearson) correlation is defined for only numeric attributes. Using correlation with one-hot encoded versions of these categorical features also makes no sense. Secondly, In classification tasks, correlation between a, let's say binary output (e.g. $0,1$), and a numeric attribute may not make much sense as it normally does in regression problems. And, when it makes sense, it tries to capture the linear relationships by definition. (e.g. when $x=y^2$ and $x$ is symmetric for example, correlation is very low or zero)
For example, let's say we have a relationship $Y=1$ only when $X>0$, else $Y=0$. If we have small negative $X$, but comparably large positive $X$, as positive $X$'s move in positive/negative direction correlation changes because of their numeric values. But, actually this feature perfectly classifies the data.
While others are not perfect heuristics, correlation might be quite misleading when assessing the utility of the feature.