I don't understand why
exit & does not work. Why does it not?
It does work.
& forks the shell, starting a new process (you could think of it as
& exit, except of course that syntax actually means something else).
exit is a shell built-in that ends the shell process -- in this case the new backgrounded shell.
> exit &  1709 > ps -p 1709 PID TTY TIME CMD + Done exit
There's your job. It's done. It worked.