I see hate like most other things, it is not inherently positive or negative. It depends on who is using it, and how it's used.
TRUE...everything depends on who is interpreting and or executing it.
We differ in that you seem to see Hate as a tool or technology/technique, and not an Emotion/Feeling.
Hate is a Feeling, Emotion or Intention to Harm or Destroy something or someone that is offensive.
When an organization is promoting an Act or Feeling the results are clear and usually unmistakable once it has been put into Action.
You have not address hate's exceptionalism that you alluded too?
Or what makes hate different for religions to advocate.?
I quote your questions below.
"Food for thought. Why does hate get the special exception?"
"What about hate makes it different than anything else a religion could advocate?"
For example, I'm an adult aged, college educated man. I can say I hate cancer. That perspective could motivate me to throw myself into eradicating cancer. My position is that's a positive.
Thats like saying "let's fight for peace"....very difficult undertaking.
True your Intent in the example above is Good or and positive, the energy of Hate is destructive and in trying to redirect it for Good purpose is creates many challenges, wherein in the end it will often end up doing more harm than good. Whilst If you were working from the Energy of Love which is Creative, the challenges would be very different and often accomplishing more good than expected.
Truism that says "the road to hell is paved with good intentions"
The long arc of any endeavor seeded with the Energy of Hate is going to be in the end destructive to all involve no matter how good the Intention.
The father of Fear is Ignorance, The child of Love is Understanding.