In Euclidean triangle geometry the Spieker point $S$ is the centroid of the triangle sides, and it is known to be the
anticomplement of the incircle center $I$ (https://faculty.evansville.edu/ck6/encyclopedia/ETC.html), which means it is in line with
$I$ and the triangle centroid $C$, such that
$$\overline{S C}~=~\frac{1}{2} \,\overline{I C}.$$
The following theorem generalizes this fact to $n$ polyhedra with inscribed spheres by a proof that is based on rescaling.
Definition:
The Spieker point of a $n$-polyhedron is the centroid of all its $n-1$ faces.
Theorem:
In every $n$-polyhedron $P$ with inscribed sphere, the incenter $I$, the centroid $C$, and the Spieker point $S$ lie in this order on a line, such that
$$\overline{S C}~=~\frac{1}{n} \,\overline{I C}.$$
Proof:
Denote by $v$ the $n$-volume, and by $a$ the area of the $n-1$-faces, and by $r$ the radius of the inscribed sphere of $P$.
For $\epsilon>0$, expand $P$ by a factor $1+\epsilon/r$ and translate it to $P_\epsilon$ such that the insphere centers of $P$ and $P_\epsilon$ coincide. Because the insphere radii are perpendicular to the faces in the touching points, the corresponding faces of $P$ and $P_\epsilon$ are parallel and have distance $\epsilon$. The volume of $P_\epsilon\backslash P$ is $ a_\epsilon ~=~\epsilon\, a +O(\epsilon^2)$, its centroid is $S_\epsilon$ and converges to $S$.
By the expansion around $I$, the centroid $C_\epsilon$ of $P_\epsilon$ lies on a line with $I$ and $C$ and
$$\overline{I C_\epsilon}~=~(1+\epsilon/r) \,\overline{I C}, $$
such that
$$\overline{C C_\epsilon}~=~\epsilon/r \,\overline{I C}. $$
On the other hand, $C_\epsilon$ is the centroid of $C$ with weight $v$ and $S_\epsilon$ with weight $ a_\epsilon$, whereby $C,C_\epsilon,$ and $S_\epsilon$ lie on a line which then also contains $I$ and
$$v\,\overline{C C_\epsilon}~=~\frac{v\,\epsilon}{r}\,\overline{I C}~=~a_\epsilon\, \overline{S_\epsilon C_\epsilon}. $$
Taking the limit $\epsilon \to 0$
results in
$$\overline{S C}~=~\frac{v}{a\,r} \,\overline{I C}.$$
Observing that the volume of the $n$-pyramid over any $n-1$ face $F$
of $P$ with tip $I$ is
$$ v(pyramid) ~=~\frac1n\,r\,area(F),$$
and that the sum $\frac1n\,a\,r$ of all these volumes is equal to the total volume $v$ of $P$, it follows that
$$\overline{S C}~=~\frac{v}{a\,r} \,\overline{I C}~=~\frac{1}{n} \,\overline{I C},$$
which
proves the theorem.
A corollary of this theorem is that in a $n$-polyhedron $P$ with inscribed sphere with centroid $\vec{c}$ and Spieker point $\vec{s}$ the incenter
is given by
$$\vec{i} ~=~ (n+1)\, \vec{c} - n\, \vec{s}.$$
As a consequence, centroid and Spieker point coincide in $n$-polyhedra with inscribed sphere if and only if incenter and centroid agree.
