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An explicit reference is Lemma 1.2.19 in T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Spaces, Vol. 1 (Springer 2016). Neither reflexivity nor separability matter here.
I haven't quite caught where the confusion arises. (The set NA$(X)$ depends of course on the norm on $X$ chosen.) And who are the authors of the book you are quoting?
Well, this does complete the proof since the metrisability of weakly compact sets in separable spaces is a well-known fact (showing the easy half of Eberlein-Shmulyan).
One needs the additional information that weakly compact sets in separable Banach spaces (like the closed linear span of the $X_n$) are metrisable; therefore one actually can extract a weakly convergent subsequence.
