No, I'm putting the long, tapered connector on the head, and the short nut on the diverter. Swapping the hose end-for-end doesn't help. Both ends of the hose have female NPT, which I can confirm because they easily thread onto another NPT fitting that I have. So the diverter is the odd thread.
The more I look into this, the more I'm thinking that the diverter is actually a "G" thread, aka BSPP (British Standard Pipe, Parallel). The fitting doesn't seem to taper, which could be why the hose won't thread onto it. Apparently BSPP is what is used in much of the rest of the world? Not sure why it would show up on my Jayco, though. I'm going to order an adapter from Amazon and try it.
One other thought - this connector was made to seal with a washer, not with thread tape. BSPP is designed to seal with a washer, but NPT is designed to seal along the threads, not with a washer. So now I'm wondering if we all have BSPP on our diverter, and for some reason I got a bad hose which has NPT instead of BSPP?