why not using this?
reg = Tetrahedron[];
Graphics3D[{Opacity[.5], Insphere[reg], reg}]
The constraints can be used to pack multiple spheres in the tetrahedron.
Frank, what is the main objective here? For example, a code like this also works, but I am not sure that is your objective:
reg = Tetrahedron[];
pnts = RandomPoint[reg, 10];
Graphics3D[{{Point[pnts]}, {Opacity[.5], reg,
MapThread[
Sphere[#1, #2] &, {pnts, Abs@SignedRegionDistance[reg, pnts]}]}}]
The objective is to pack the spheres in the tetrahedron as tightly as possible without overlap
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