They wouldn't need to eliminate Israel to get them to consider negotiating over Palestine, they didn't need to actually take the Sinai to get Israel to negotiate over that, but they did have to invade it. The rhetoric required to rally support for the war couldn't have said that they wanted to take a little to get a place to start negotiating from. Israel was actively colonizing the Sinai just like Gaza and the West Bank, if Egypt hadn't started the war around that time, Israel would have quickly considered the settlements of the Sinai as vital as the settlements in Gaza and the West Bank. Today it might seem obvious that Israel would trade that land for recognition, or that they'd have focused mainly on the West Bank to settle, but at the time it wasn't.
I don't think they were even trying to get close to the 1967 borders, that would carry too much risk of Israel using one of their rumored nuclear weapons. This isn't the kind of thing that can be used in speeches to rally the troops.