Doesn't Anchoring Vignettes merely move the problem of coming up with DIF-free survey questions back one level (from self-assessments to vignettes), and so in the end you have the same problem?

No, the goal of survey design under this approach is not to design DIF-free vignette questions (which is as difficult or impossible as for self-assessment questions). The approach allows respondents to interpret vignette questions in completely different ways. Instead, the goal of survey design is to write vignette questions that have the same types of DIF as the self-assessments, since that provides the necessary information with which we can measure DIF, and with that we can then correct the self-assessments. Since the same respondent will be interpreting both the vignettes and the self-assessment questions, the assumptions of the technique are much more likely to be met than having to to design DIF-free questions.