> All formalisations of non-classical logic operate in the framework of first-order logic, in the sense that the informal meta-language in which the non-classical logics are explained in is traditional first-order logic

What gave you this idea? It is frequently useful to work over different base logics to obtain models of non-classical logics. E.g., when you work in categorical logic you usually work over an intuitionistic logic which can then be interpreted in an arbitrary topos. This gives you the ability to internalize any constructions you do "on the outside".

There is a large community of mathematicians fixated on classical first-order logic, but this is just because of tradition. For models of classical first-order set theory it just doesn't make much of a difference. This is not true of "all formalizations of non-classical logic".