There cannot be a meaningful definition of a centroid for all sets of points on $S^n\subset\mathbb R^{n+1}$ that is invariant under isometries. To see this, consider the corners $p_0$, \dots, $p_{n+1}\in S^n$ of a regular $(n+2)$-simplex. There are isometries of $S^n$ that permute the corners in any possible way, and these form a representation of the symmetric group $S_{n+2}$. If $p\in S^n$ was the centroid and $\gamma\in S_{n+2}$ one of the isometries that fix $\{p_0,\dots,p_{n+1}\}$ as a set, then $\gamma(x)$ would also be a legitimate centroid. But the action of $S_{n+2}$ has no common fixpoint.

There cannot be a meaningful definition of a unique centroid for all sets of points on $S^n\subset\mathbb R^{n+1}$ that is invariant under isometries. To see this, consider the corners $p_0, \dots, p_{n+1}\in S^n$ of a regular $(n+2)$-simplex. There are isometries of $S^n$ that permute the corners in any possible way, and these form a representation of the symmetric group $\Sigma_{n+2}$. If $p\in S^n$ was the centroid and $\gamma\in\Sigma_{n+2}$ one of the isometries that fix $\{p_0,\dots,p_{n+1}\}$ as a set, then $\gamma(p)$ would also be a legitimate centroid. But the action of $S_{n+2}$ has no common fixpoint.