It's a logistic regression model, a basic statistical technique which wouldn't have even come under "machine learning" a few years ago. Later they use some kind of LASSO regression to penalise the inclusion of redundant features.

"Combing through the correlations", it seems, literally means calculating the (Pearson) correlation between two variables (success vs. an input feature) and adding some interpretation, as they do on p6 of the arXiv paper. For test/training data, it looks like they just used a 30/70% split rather than k-fold cross-validation and holdout, but I'm sure it makes no difference either way and in this case (as often) is trivial to design and implement. Presumably their AUROC could be increased just by dropping in an SVM or a Random Forest in place of the logistic regression.

From what I've skimmed of the paper you're over-estimating the complexity of the study.