Re: Is it kokyu-nage if you don't use kokyu-power?
I would say no.
Now, if it were just a poorly executed kokyu nage by a student who said "let me show you my kokyu nage" and there was no real "kokyu", I wouldn't say to him "that was no koky nage". I might try to offer some pointers.
On the other hand, is some karate teacher says "I'm an Aikido master too" and came up with some lame excuse for kokyu nage, I would then say that was not Kokyu Nage. Then of course his response would likely be that I was just being political.
I had a similar situation when a Tae Kwon Do teacher told me they do "Aikido techniques" as part of thier self defense program, then showed me some very crude, poorly performed half karate half jujutsu technique. He had his "uke" step into a front stance and hang out his fist as if punching his face. He did a hard "rising block", punched him back and twisted his wrist as he fell to the ground.
My question was "what makes that an Aikido technique?". He said it was Kote Gaeshi and that is found in all Aikido.
So the next question again was "where's the Aiki that would make it Aikido?".
You can also find the same situation with people who teach Jujutsu and call it Aikido. Sure it looks like Aikido to some, but it certainly doesn't feel like Aikido to someone who has experienced Aikido.
The same would go for a kokyu-less kokyu nage. It just is not kokyu nage.