Dedekind cuts can similarly by recast as a function that gives -1,0, or 1 if a number is below, the same, or above the real number. This would be a great improvement over the usual presentation.
But beyond that difference, consider the solution to f(x) = 0 in terms of the intermediate value theorem for an increasing function. The Dedekind cut solution is A = {x | f(x) < 0}. That is the answer. From the perspective of Dedekind cuts, there is nothing further to compute and it has provided nothing to compute with.
The oracle answer would be the rule that says yes to a<b if f(a)*f(b) <=0. This gives a direct mechanism for testing intervals and the idea of narrowing intervals by picking a point inside a Yes interval is very natural from this perspective. I do not see that suggestion popping out of the Dedekind cut approach.
The rational interval approach specifically is targeting narrowing solutions to the answer with error bounds built into it. If I tell you that 3 is in the set A, you have no intrinsic sense of how good of an approximation that is. But if I tell you that the target is in the interval 3 to 3.1, then you have a sense of the precision. Oracles promote having accuracy as foundational.