The authors did not say anything like "$\mathbb R^{\mathbb N}$ is not high-dimensional enough." 

Rather, they said 

>"finite dimensional convergence [...] is too weak a result to develop genuinely high-dimensional inference methods". 

Indeed, if one wants to study the behavior of $S_{n,p}:=S_n$ for large $n$ and $p$, then it is not enough to know the behavior of $S_{n,p}$ for large $n$ but only for a fixed finite set of values $p$. You already "understand that weak convergence (= convergence in distribution) in $\mathbb R^{\mathbb N}$ is equivalent to weak convergence of the finite dimensional marginals." So, the highlighted thesis follows.