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The second comment of him how to obtain the stronger statement is basically that his proof can be adapted to the more general case, which I was aware of. I'm lacking a reference for exactly this generalization for readers that are not experienced with the work.
The first "trick" he mentions there ($X\simeq X_0\times\pi_0(X)$ and both factors are $n$-fold loop spaces) to get the result is not satisfying for me as a want the weak equivalence to be a map of $\mathcal{C}_n$-spaces.
