Well you can see from Theorem 2 that the ADM 4-momentum is parallel to the time-like killing vector at spatial infinity with the proportionality factor containing the Komar mass itself, so indeed the ADM 3-momentum vanishes. What this means is that the rest frame defined by the ADM 4-momentum agrees with the rest frame defined by the time-like killing vector at spatial infinity, a result which is by no means obvious to me from the definition of the ADM 4-momentum as evaluated in a stationary space-time; in particular, the fact that the twist ##\omega_a = \epsilon_{abcd}\xi^b \nabla^c \xi^d## of the time-like killing field ##\xi^a## satisfies ##\lim_{\rightarrow i^0}\hat{\omega}_a = 0##, which was a crucial part of the proof, is not obvious to me just from physical intuition (the hat indicates the twist associated with ##\hat{\nabla}_a##, ##\hat{\epsilon}_{abcd}##, and ##\hat{\xi^a}##). Indeed this result (that is, including the part about the proportionality factor containing the Komar mass) is only obtained in retrospect after the proof of Lemma 2.3.
Of course even before the proof of Lemma 2.3, one could argue through physical intuition that for a stationary space-time, the ADM 3-momentum must clearly vanish but just because the ADM 3-momentum vanishes doesn't mean a priori that the ADM energy must equal the Komar energy; the conclusion is non-trivial as Lemma 2.3 shows.