The purpose of those trips was for Zuko to establish trust between them, though. Aang, Katara, and Sokka were all deeply involved with Zuko from the very first episode, and he had been hunting them specifically. Toph just showed up later and didn't have the same connections. She didn't even have the same connections that the others did to firebenders in general, since they wiped out all Air Nomads and decimated the Water Tribes, especially the Southern one. The other three all had deep-seated personal issues with both Zuko as Zuko and with Zuko as a Fire Nation prince, and those trips established that he was not who he was before and was indeed an ally. Toph didn't really need that.