RegionWithin does not evaluate for a sphere in a tetrahedron, so it was necessary to use a different approach.
First find the faces of the tetrahedron
In[]:=
faces=PolyhedronData["Tetrahedron","Faces","Polygon"];
Next find the vertices of the faces
In[]:=
verts=PolygonCoordinates/@faces;
Then generate infinite planes containing the faces
In[]:=
planes=InfinitePlane/@verts;
And then calculate the equations of the infinite planes
In[]:=
planeEqs=RegionConvert[#,"Implicit"][[1]]/.x.->xt,y.->yt,z.->zt&/@planes;
Calculate the square of the minimum distance from a point to the planes, using the plane equations
In[]:=
mins=MinValue[{
2
(xt-x)
+
2
(yt-y)
+
2
(zt-z)
,#},{xt,yt,zt}]&/@planeEqs;
Constrain the point to be farther from the planes than the radius of the sphere.
In[]:=
cons=FullSimplify[Thread[
2
r
<=mins]];
It is also necessary to confine the point to inside the tetrahedron
In[]:=
rw=List@@RegionWithin[Tetrahedron[],Point[{x,y,z}]];
Combine the two kinds of constraints
In[]:=
allcons=Join[cons,rw];
Find the largest sphere that will fit inside the tetrahedron
In[]:=
sln=Maximize[{r,allcons},{r,x,y,z}]
Out[]=

1
2
6
,r
1
2
6
,x0,y0,z0
Show the result
In[]:=
Show[​​Graphics3D@{Opacity[0.3],Tetrahedron[]},​​Graphics3D@Ball[{0,0,0},sln[[1]]]​​,ImageSize->Small,Axes->True]
Out[]=