Why is not AUC or other metrics used for splitting nodes in decision trees?
On accuracy:
Why we use information gain over accuracy as splitting criterion in decision tree?

AUC has been explored; it seems to work well, but is slower:
https://www.semanticscholar.org/paper/Learning-Decision-Trees-Using-the-Area-Under-the-Ferri-Flach/46e40f487e555277033f188778d6c5c05df8daa4
http://proceedings.mlr.press/v7/doetsch09.html