How does 'Agent Doe' end up where he does in Predestination?

ref : Predestination Explained

Barkeep has been basically chasing his own self from a future time. The time leaps have caused Barkeep to grow maniacal and turn into the Fizzle bomber.

Barkeep does not report the time device which he’s unable to decommission. This indicates that he continues to use the device; time jumps are indicated to have an effect on the mind. Let’s not forget about the fact that every other event in this story is predestined to happen. This should be no exception to the rule. It only makes sense that Barkeep grows old to become the insane Fizzle bomber.

Though the temporal agent has been disarming bombs, he doesn't know that his future self is the one planting them. This is why they refer to it as a snake eating its own tail.

Finally at the laundromat .. the barkeep comes to know that the fizzle bomber is merely an older version of himself. All this time he has loathed the fizzle bomber only to find out that it is his future self. While the fizzle bomber says that killing him is what will cause the barkeep to start a journey that ends up with him being the fizzle bomber .. the barkeep is unable to see past his hatred for the fizzle bomber (even though it is his future self). In his hated and anger he kills his future self.

After this, we are shown that his machine doesn't deactivate. He doesn't report it either. At this point the barkeep is not insane. He going to start using the time machine to prevent himself from becoming the fizzle bomber. But the very act of him doing so will result in his insanity and hence become the fizzle bomber years later.