I don't know Toyota, but usually the neutral safety switch does not permit current to the starter relay / solonoid unless the trans is in either neutral or park. The same switch does both.
On most cars today, the neutral safety switch is more than that, it is also the transmission manual level range selector sensor (MLPS) and it tells the computer which gear you have selected with the shift lever.
My limited experience with Japanese designs tells me that you should also be looking the the ignition switch as well as the starter relay (starter solonoid, piggy back on the starter).
Good luck.
