Electroweak penguin $B$ decays with neutrinos or $\tau$ leptons in the final state are sensitive probes of physics beyond the Standard Model. Using the full Belle and Run 1 Belle II datasets, the first search for $B^0 \to K^0_S\tau^+\tau^-$ sets the upper limit on the branching fraction to $\mathcal{B} < 8.3 \cdot 10^{-4}$, the search for $B^+ \to K^+\tau^+\tau^-$ sets $\mathcal{B} < 5.6 \cdot 10^{-4}$, and their isospin combination gives ${\mathcal{B}(B \to K\tau^+\tau^-) < 5.4 \cdot 10^{-4}}$, all at 90\% confidence level (CL$_s$). The first search for inclusive $B \to X_s\nu\bar\nu$ decays at Belle II sets $\mathcal{B} < 3.3 \cdot 10^{-4}$. Finally, the Belle II $B^+ \to K^+\nu\bar\nu$ result is reinterpreted in the Weak Effective Theory using its published likelihood. The data prefer an enhanced vector contribution and mildly favour a nonzero tensor contribution.