I'm inclined to agre with Deb on this one, I don't think Jon records songs now that his voice can't handle. Actually he never really did, when he recorded Prayer he could hit the high notes on it.
As for the box set I really think that those songs got left off records at the time because A) they didn't fit in with the theme of the album being recorded, B) Jon and the band didn't think they were good enough to make the cut at the time, or C) Jon just flat out didn't like the song and didn't want to do it.