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 $x^\alpha_n$ in such a way that $x^\alpha_{\alpha(k)}$ is not convergent. So there will be no common reindexing for this family of sequences.

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. Perhaps this cardinal will be provably equal to one of the other better known cardinal characteristics.