Renovating basement, keep original studs or do this instead?

I don't have time to watch this episode right now, but I assume that they got rid of the moisture first, and then put in the polystyrene. I would not put anything new up until the problem is fixed. Polystyrene will help since it is not affected by water, but if moisture gets into the house it is going to give you problems. If your original studs are right up next to the exterior wall then it would be best if you could put the foam behind it and get rid of heat transfer from the concrete and wood. I would also use a treated base plate for your wall that will further protect you (I'm sure Tom did that too--it's usually code). I just don't want you to have all new walls completed the way the show states and still have moisture problems. If you can't get rid of the water problem yourself by inspecting eave troughs and downspouts above to make sure water flows away from the foundation, making sure all landscaping next to the house slopes away for water run off. purchasing a dehumidifier for the basement that will help if inadequate duct work is installed in the area (like I just had to do when I found mold beginning to grow in the corner of a basement room where water was seeping down through the wall because my down spout was not fastened correctly and the room has no air circulation--thus the stagnant odor that many basements will have).