It has it's flaws, but watch some of the explainers I linked below, and give it another go. I did enjoy it better the second time.
Yes, it's implied that Washington created the Tenet organization in the future. As to why he has to test himself, good question, and you can take your pick.
This seems to be a standard test applied to ALL CIA agents as a recruiting tool for Tenet. If you're dedicated enough to provably try to kill yourself to protect knowledge that would get other people killed, then you've passed the entrance exam. At this point in the past, the older Washington isn't around, so his younger self is just recruited like everyone else is. It appears that no one, except Neil, even knows Washington is the founder of Tenet in the future.
Since the movie implies that the past cannot be changed, then his recruitment this way already happened and was inevitable.
- "The same idiots whining about cancel culture are the people who tried to cancel pretty much every video game, book genre, and RPG when I was a kid." - James S. A. Corey