I've read one proof, rather long, in Allen Hatcher's book. There the key is Lemma 4.70, which uses the relative Hurewicz Theorem. But there is another, shorter proof in J.P.May's book "A concise course in algebraic topology", chapter 22.4. My problem with that proof is that I can't find any way that it uses simplicity of the space. We discussed this argument as proof for general CW-complexes on a seminar, and could not find an error.

I can write that part of the proof, which is omitted in the text, here: Let B $\subseteq C\left(\alpha_n\right)$ be the image of $X \times [0,2/3)$, A is the image of $X \times (1/3,1] \cup X_n$. Here $A$ and $B$ are as in the definition of an excisive triad.

Then if $C=A \cap B$, there are homotopy equivalences of pairs $(A,C) \sim \left(M\left(\alpha_n\right),X\right)$ and $\left(B,C\right) \sim \left(CX,X\right)$. Also, if we identify $\pi_k\left(A,C\right) \simeq \pi_k\left(M\left(\alpha_n\right),X\right) \simeq \pi_{k-1}\left(F\left(X \to M\left(\alpha_n \right) \right)\right) \simeq \pi_{k-1}\left(F\left(X \to X_n\right) \right)$. The only nontrivial identification here is by definition of relative homotopy groups.

Moreover, this identification commutes with morphisms to $\pi_{k-1}\left(C\left(\alpha_n\right)\right)$. The first of these is $\pi_{k-1}(\eta) : \pi_{k-1}\left(F\left(X \to X_n\right) \right) \to \pi_k\left(C\left(\alpha_n\right)\right)$ and the homomorphism induced by the morphism of pairs $(A,C) \to \left(C\left(\alpha_n\right),B\right)$ : $\pi_k\left(A,C\right) \to \pi_k\left(C\left(\alpha_n\right),B\right) \simeq \pi_k\left(C\left(\alpha_n\right),\ast \right)$.

This commutativity is evident if we choose the projection as the homotopy equivalence $M(\alpha_n) \to X_n$ and retraction along $I$ as the deformation retraction of $B$ to $\ast$. So, we only need to check that the morphisms induced by $(A,C) \to \left(C\left(\alpha_n\right),B\right)$ are isomorphisms for k up to n+2.

This is true by the Homotopy excision theorem, applied to $A,B$ with $n_{ex}=2, m_{ex}=n+2$. The theorem applies because of homotopy long exact sequences of pairs which say $\pi_l(CX,X) \simeq \pi_{l-1}(X)$, so $(B,C)$ is $1$-connected, which is $n_{ex}-1$-connected, so $n_{ex}=2$. The other exact sequence is $$ \pi_l(X) \to \pi_l(X_n) \to \pi_l(A,C) \to \pi_{l-1}(X) \to \pi_{l-1}(X_n).$$ Here the first arrow is an epimorphism for all $l$ and the last is a monomorphism for $l-1 \leq n$. So, $\pi_l(A,C)=0$ if $l \leq n+1$. This means that $(A,C)$ is $(n+1)$ - connected, so $m_{ex}=n+2$. Thus, the theorem applies.