Transfer Game Data between iPhones (same Apple account)

I have an iPhone 5 and an iPhone 6s.

I got the 6s while on vacation and couldn't do a backup from iTunes, so I set it up as a new phone.

But, by the time I got home, I had so much new stuff on the new phone, that I didn't want to wipe it in order to put on my old stuff (and I only wanted about a third of my old stuff anyway).

So... as far as I can tell, I cannot access my old game data from the six games I'd like to continue to play on my new device unless I wipe my new phone and bring over everything - 2/3rds of which I don't want on my new phone.

Is that right?

Why is that? Why on earth can't I move data files from one of my Apple devices to another, with the same Apple and Game Center accounts? I don't understand that restriction at all!

Basically I'd like to start with a clean device and install just a few of my old apps (with my old progress).

Am I missing something? Is it doable? And if not... why on earth is it not allowed??? That seems crazy to me!