IPS upper bound for subset sum axiom
First, Kaveh is correct that the verification for IPS is randomized, so all it would show is $\mathsf{NP} \subseteq \mathsf{coAM}$ (not $\mathsf{NP} = \mathsf{coNP}$). However, this alone would still be enough to collapse the polynomial hierarchy.

Second, I think the actual thing you are missing here is that the IPS proofs they give have size polynomial in the size of $|A| = |\{\sum_i \alpha_i x_i : \vec{x} \in \{0,1\}^n\}|$, which can be exponential in $n$, in general.