On the proof techniques of Udi Manber
It is not a rigorous answer, but if we are to posit in the way of Udi Manber using induction, we are tacitly positing a recursively enumerable algorithm that can terminate. Udi Manber's algorithmic approach though does not necessarily posit a primitive recursive algorithm. If the problem in our class is not recursively enumerable, then we cannot use Udi Manber's approach, and it may as well be uncomputable. A primitive recursive algorithm is likely to be more efficient than a recursively enumerable one.