Further unqualified, but highly experienced, opinion.
I get Classic Migraine too. They started about 16 years ago, so I know the difference (and have been used as a Guinea Pig because of it).
LR, you mention blood supply. My theory of cold is this.
I saw a documentary on the evolution of Man. At some point we developed a pair of holes in our temples, and scientists were puzzled as to their pupose. Dopplers showed that the blood flow could be in either direction, and they concluded that it was a brain cooling system. My guess was that a CH sufferer had a screwed system, and that fooling the nerves responsible for flow control is what stopped whatever other nerve was causing the pain from reacting.
Not a foolproof thesis, not least because CH is partly defined by it's one sided presentation. However, that's not to say that only one flow control is malfunctioning.
Give me migraine any day.