The traditional way of teaching dynamic programming worked well in the traditional CS curriculum that had a lot more mathematics than today. It leveraged what the students had already learned at that point.

Take the definition of the problem. Transform it into an equivalent definition that looks like the inductive step in a proof by induction. Look at the partial order defined by the dependencies between the subproblems in the inductive step. Solve the subproblems in an order that is consistent with the partial order. (For bonus points, find an order that is asymptotically faster in easy situations, such as when the edit distance is small. You may have to rule out paths that cannot be part of an optimal solution.)

That's first or second year mathematics in the traditional curriculum.

Today, and actually since the 90s or maybe even earlier, many CS students have no particular interest in mathematics. Classes that rely on mathematics have to spend less time on the actual content, as they have to cover the prerequisites and find alternate ways of explaining things.