Theorem 1 of Milnor's paper "A procedure for killing homotopy groups of differentiable manifolds" states that two manifolds are in the same cobordism class if and only if they can be obtained from each other by a sequence of surgeries (meaning they are $\chi$-equivalent, following Milnor's terminology).

In his proof that cobordism implies $\chi$-equivalence, he utilizes a Morse function on the cobordism manifold, although not explicitly. Is there a reference containing a more modern reformulation of this proof via Morse functions? I would also like to avoid introducing the language of attaching handles if possible.