There is a counterexample $\mathcal{A}$ of size continuum. To see this, observe that there are only continuum many reindexing functions $\alpha:k\mapsto \alpha(k)$, and for each such $\alpha$, we may pick a sequence $\{x^\alpha_n\}_n$ in such a way that $\{x^\alpha_{\alpha(k)}\}_k$ is not convergent. That is, we define the $\alpha$th sequence specifically so that $\alpha$ does not work with it. Because of this, there can be no common reindexing $\alpha$ for this family of sequences, since every $\alpha$ fails with the $\alpha$th sequence $\{x^\alpha_n\}_n$.
In this case that CH fails, one might hope for a smaller uncountable counterexample, and we could introduce a cardinal characteristic here, for the smallest size of a set $\mathcal{A}$ admitting no such common reindexing. Let us call it the common convergence number, denoted 𝕔. Perhaps this cardinal will be provably equal to one of the other better known cardinal characteristics. The fact that subsequences of convergent sequences are also convergent seems to suggest that it could be consistent with ZFC that the smallest counterexample could be less than continuum.
In light of the counterexample, we seem led naturally to the question:
Question. How large is the smallest counterexample? In other words, how big is 𝕔?
It makes sense to me to consider 𝕔 as defined with respect to real sequences only, but I am unsure whether there is any dependence on the underlying space.
(I encourage others to post further answers about this on this same question thread.)