After several schools of the philosophy of mathematics met limitations one after the other in the 20th century, the assumption that any foundation could be successfully modelled within mathematics itself began to be heavily challenged.
Today one refers more ambiguously to the foundations of mathematics to avoid giving the impression that there is a 'problem' that can be solved in the sense of a science or mathematics problem, with a single right answer that is checked by means describable in proof theory.
The term 'foundations problem' only occurs in literature that makes the assumption that there is such a provable and single foundation. In the 1970s, many arguing the prevalence of quasi-empiricism in mathematics denied that assumption - and sought to focus on mathematical practice and later a cognitive science of mathematics - an outright empirical basis for why mathematics works, and "Where Mathematics Comes From".