The Higgs mechanism has been understood in the framework of quantum field theory since the very beginning i.e. since the 1960s. Quantum field theory may be constructed as a quantization of its classical limit, i.e. of a classical field theory, so that's what's important for understanding some basic properties of the Higgs mechanism, too. But many other things, such as the existence of the Higgs boson itself, require one to study the quantum field theory, not just the simpler classical field theory. Classical field theory doesn't really imply the existence of any particles. To have particles described by fields, one usually needs quantum field theory, too.
The Higgs mechanism is a mechanism in quantum gauge (field) theory using a scalar field whose vacuum expectation value breaks a gauge symmetry which makes some gauge bosons massive. By definition, the Higgs mechanism is inseparable from quantum field theories and from gauge theories. Everything that is called "Higgs something" in physics is inseparable from gauge theories (a subset of quantum field theories), too.
The importance of all these things in particle physics (and even condensed matter physics, sort of) is very high.
Concerning the Wikipedia page: in condensed matter physics, the objects that break the symmetry are often composite – Cooper pairs (of electrons) in superconductivity. All available evidence in particle physics indicates that the field breaking the electroweak symmetry of the Standard Model is elementary, not composite. Some substructure may ultimately be found but there is absolutely nothing wrong or incomplete about elementary scalar fields – at the level of quantum field theory – and the Wikipedia page sentence was entirely incorrect.