I always felt FATE was a representation of humanity. Wanting to be good, but failing. When it said the whole," I love you so much.." bit it showed how FATE, despite being a robot, had become the embodyment of the humans it saw around it. He was confused between love and hate and didn't know if it should kill them or let them live. It didn't even know if letting them live was good because it saw how humans hurt one another. Robo did the same thing around Crono's group. However, Robo saw the better side of humanity throughout his life and tried to follow it. Both FATE and Robo gained emotions by being with humans. FATE just had the misfortune of seeing so many lost humans over the years that FATE itself got lost.
It most likely did want to start its own race one day, but it couldn't because there was nothing that it could model it after. After all look at those humans with their love and their hate and how much of a paradox they are to themselves. When it looked into the flame, it looked into itself and saw it had become like the humans. It was so confused with what it had become that its actions could no longer be explained.