I fixed the "inverted switch" problem in the original wiring diagram... bastards.
http://medinaline.net/pics/megastrat-wiring-fixed.jpg
I am trying to get up the nerve to get started on it right now. Wish me luck.
----------- Update: 8:35PM -------------
I installed the new 5 way switch... Unfortunately, the behavior is exactly the same. Only the rhythm pickup works and only with the push/pull switch down. Time to replace the push/pull switch now. Wish me more luck than I got last time.
----------- Update: 9:32PM -------------
OK, I feel generally better about things. The situation is exactly the same now that I replaced the push/pull knob. I have a feeling there is either something fundamental I don't understand about the wiring diagram, or there is something fundamentally wrong with the wiring diagram. Either way, I think I can rule out bad soldering as the problem :)
Time to go pay someone to fix it.