I figured it out and this isn't a bug. It is intended behavior. There is literally a comment in the game files describing it as an easter egg. Go to Hearts of Iron IV\events\BFTB_Turkey.txt (not Turkey.txt) and look for event bftb_turkey.24 option bftb_turkey.24.j. Note that the condition is not doing "ratify the six arrows."
This isn't a very useful exploit though. If you don't do the Ottoman focus tree you'll miss out on some important bonuses and, in particular, this version of the Sultan won't die and let you get his awesome daughter. What's not clear to me from the game files is if you can still go down the focus tree, replace the Sultan with a Democratic leader, and then have a civil war to re-restore the Sultan. But even if you can do this, it is probably not worth delaying all those bonuses just to get the military ones first.