The point is measuring actual student skill, not busywork throughput or the ability to meet arbitrary deadlines. That's fine, IMHO.
But we still need an objective way to measure the skills - e.g. an exam, an in-class assignment or presentation, something like that. Students should be allowed to retake exams if they fail. But then, if students from affluent families do better (because they've got more time to study!), they should be given a better grade, otherwise things lose meaning. Maybe we could switch to performance-based learning, where students graduate later, but in the end they achieve a certain skill level.
The improvement should come from changing poor students' environment - try giving them enough room to study and the right tools - not by dropping the grade system.