But if the "burden" consists of the priest hearing the person's confession and telling him in the privacy of the confessional that his first marriage is annuled, then I believe a LOT of priests today would be happy to do so. Such a truncated annulment process would provide the fig leaf they could hide behind, and allow them to issue "annulments" with even more wild abandon than they already do.
The irony is, though, as another poster has already pointed out, that many, if not most of the "marriages" contracted by Catholics today may in fact not be sacramental marriages at all because of the lack of understanding of the requirements of the relationship into which they are entering.