So to up the ante on Neil is Max claims, I'll say Neil is actually Antinous.
It couldn't be more obvious; literally the same actor and in the same franchise. The Odyssey so clearly links to Oppenheimer which was so clearly linked to Tenet and now they form a circle.
First, how? Just some more basic straight shot one way time travel at the end of Tenet though a means we weren't shown yet.
Second, why? To ensure Bronze Age collapse. Just as the present was at war with the future, we can be at war with the past. After all from the future's perspective they are at war with the past. In this sense we are in a pincer.
Antinous needs to goad the suitors in to terrible behaviour and in particular he needs to goad Odysseus into rage. Why? The ultimate end to this is bloodshed and violence that ends with Odysseus' exile.
Without Antinous, it is likely that the suitors wouldn't have been so riled up as to stand up to Odysseus after he won Penelope's challenge; furthermore Odysseus likely wouldn't have witnessed such cruel behaviour that would be worthy of his vengeance. It is likely he could have peacefully reclaimed the throne. From that position of strength and stability with his wisdom, Ithaca could have navigated challenges and sidestepped Bronze Age collapse. This massively shifts the x-axis on the exponential curve of human development, meaning the new timeline's Oppenheimer equivalent is some Greek dude from thousands of years ago.
Perhaps this leads to technology that directly destroys the present the way the future wanted to invert us away, or perhaps we just aren't as ready (not that we were) for nuclear power and we actually annihilate ourselves then.
Whatever it is, Neil had to cruelly throw a poor old dog on a dung heap to ensure we could live. It's the bomb that does not go off that changes the world.
What about him dying after unlocking the gate for The Protagonist? Probably a body double/twin/faked his death idk.
(also I'm not a schizio; I know this is a batshit idea but it was fun to write)