These methods use neural nets or trees as their core framework. I'm wondering if there are any simpler methodologies available i.e. the linear regression (OLS) for predictive ranking?
The most appropriate variation of regression is ordinal regression. Ordinal regression tends to not perform well because of the complexity and sparseness of most ranking problems.
Almost all supervised learning algorithms have a learning to rank variation. Typically, learning to rank problems involve text so algorithms that work well with text. Ranking support vector machines (SVM) was a popular choice before the deep learning revolution.