We’ll justify $3 \rightarrow 1$ by the contrapositive. Fix $s \in \prod A_i.$ Identify $\bigsqcup A_i \setminus \{s_j: j<\omega\}$ with $\{t \in \prod A_i: \exists ! j (s_j \neq t_j)\}.$ The latter injects into $\mathbb{R}$ and is thus orderable.
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