You have found the Extra Special Behind the Scenes Content for And If Thou Wilt, Forget. This is not necessary information to have in order to enjoy the fic; it's just fun bonus lore.
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In the S5 Q&A of TMA (boy, that's a lot of abbreviations), Jonny stated that a necessary aspect of becoming an Avatar is death of some kind. He also added that it needn't be literal.
Can you pinpoint the point in Forget where Tim Became?
It was the moment he decided not to go to Great Yarmouth.
If you've ever read the book Little People by Tom Holt, you'll already be familiar with this concept. The premise of that book is a bit convoluted, but basically, it hinges on the fact that for every human being in our world, there is a corresponding elf in the elven realm. (The exception to this is the main character, whose father was an elf and whose mother was human, so he is himself on both sides of the line and essentially has dual citizenship.) There is very nearly an apocalypse precipitated by the fact that the MC's human girlfriend has met her elf version and the two can't exist in the same space at the same time without ending the world. In the end - spoiler alert - the main character solves the problem by hitting on a very simple solution: He breaks his girlfriend's heart, utterly destroys her love for him, and because what makes her and the elf version the same is their unconditional love for the main character, they are no longer the same person.
It's the same sort of thing here. One of the fundamental things that makes Tim Tim - one of the driving forces in his having joined the Institute and fought to stop the Unknowing - is wanting revenge for his brother. Wanting vengeance for Danny.
Tim had a choice to make. He could get that vengeance, but at the cost of the Archivist's safety. This would be his one and only shot to stop Jonah Magnus. The likelihood was that he would die, either blowing up the Unknowing or at Jonah's hands, because he couldn't keep the secret much longer. And in a world where he stayed Tim, he chose to go to Great Yarmouth, maybe trust Gerry to stop Jonah, but either way go to the House of Wax.
However. He chose this route. He chose to stop Jonah, to protect the Archivist. He chose to let his brother go. Maybe he even recognized that there wouldn't even be that much vengeance to be had, or more than momentary catharsis. Whatever the core of it, it doesn't matter.
Tim, symbolically, died when he gave up what made him Tim - the urge to avenge his brother.
That's why Oliver didn't need to give him the choice between moving on or coming back. There was no dying for him at that point. Tim might have had an actual choice, but the Guardian had to do what the Archivist did, and since Jon chose to come back, there was never any question that Tim would.
He had already Become.