We can assume that the set $\{p_1,\dots,p_n\}$ is maximal. Note that the set $$K_{n,n-1}=\{\,x\in M\mid |x-p_i|\ge\tfrac\pi2\ \text{for}\ i<n-1\,\}$$ is convex and it contains a point $z$ such that $|p_i-z|=\tfrac\pi2$ for any $i<n$. Since $\{p_1,\dots,p_n\}$ is maximal, $|p_n-z|\le \tfrac\pi2$

Flow $z$ in the gradient flow for $\mathrm{dist}_{p_{n-1}}$ for short time. We get a point $z'$ which will satisfy all your conditions for $\eta_{n,n-1}$.

We can assume that the set $\{p_1,\dots,p_n\}$ is maximal. Note that the set $$K_{n,n-1}=\{\,x\in M\mid |x-p_i|\ge\tfrac\pi2\ \text{for}\ i<n-1\,\}$$ is convex and it contains a point $z$ such that $|p_i-z|=\tfrac\pi2$ for any $i<n$.

Since $\{p_1,\dots,p_n\}$ is maximal, $|p_n-z|\le \tfrac\pi2$. By comparison, we get $$\measuredangle[z\,^{p_i}_{p_j}]>\tfrac\pi2$$ for all $i\ne j$.

Flow $z$ in the gradient flow for $\mathrm{dist}_{p_{n-1}}$ in $K_{n,n-1}$ for short time. We get a point $z'$ which will satisfy all your conditions for $\eta_{n,n-1}$.