It's a very informative site and I really enjoy reading its. But so far, it seems to me that I could not use these to help my students understand this problem. Do you know any simpler and more understandable method?
Do I understand you correctly - are you looking for a way to just calculate the 0.74 value?
Take a look at the lattice - it is made of repeated identical regular tetrahedrons. You can easily calculate its volume, you should also have no problems with calculating what part of this tehrahedron volume is taken by the spheres located in vertices. That should be a starting point.
Take a look at the picture - it is only 2-D case, but hopefully you should get what I mean. Blue surface is tehrahedron minus red. Now do it in 3d.