Yes, all education should be competency-based, because there is no point to leaving a child in a grade if they have already mastered the material. Psychologists completely overthink the social aspects of school, as if life is perfect as long as you spend your days with your age peers. If a student knows, they should move up.
I think education would be much more efficient if it was competency-based. I also believe it would be much easier for students in the younger grades to master one set of competency skills at a time rather than jumping between six or seven topics like we currently do in education. Competency-based learning seems like a win-win situation for everyone.
Education should not be competency based, simply because there are certain skills, forms of expertise, and what we now call competencies that can not be measured. Emotional intelligence is a concept now beginning to come to the fore that shows the short comings of models of education based on "competency."
I do not believe all education should be competency-based. The reason I say this is because it is virtually impossible to show competency in all fields. Yes, students should be tested for understanding, but that does not mean it is 100% accurate. For things like art, music, and physical education it is much more difficult to really test for competency.