The reality in my experience (as a university lecturer) is that the burden of proof very much falls on the accuser rather than the accused.

How is that? Students already are required to submit a lot of evidence with their work such as research notes, plans, lab work etc. to prove they’ve actually done that and if plagiarism detection systems flag any of their work they have to defend it rather than the institution having to investigate and build evidence on its own outside of w/e shoddy plagiarism detection system they bought told them.

What could that burden of proof look like?

Not sure how a student could prove that the negative, apart from videotaping themselves writing the essay, or only being allowed type it on an airgapped machine owned by the university.

Doesn't feel very feasible to implement.

and if the student is not a complete tool, he can definitely learn the output and replicate it very, very closely

Even before GPT, schools should require students to submit derailed outlines and notes of their work. For one, students need to learn these skills. For two, it helps protect against more plagiarism and cheating.