It appears to me that the advertising was based on an early draft of the script and not what eventually ended up on film.
Which is of course nonsense, but it still looks like it.
I do think that this John Harrison thing was a last minute addition when they cast Cumberbatch. Orci & Co got to know in the middle of pre-production that everybody else already assumed they were going for Khan. So they tried to keep it a mystery as long as possible. So they cast Cumberbatch instead of Del Toro, and rewrote the script around him. Which eventually resulted in a character reveal that isn't spectacular, or even necessary. Would the film really be different with John Harrison being JUST John Harrison?
In fact, the character and the plot around him is so ambigous, it seems they couldn't decide between Khan and Gary Mitchell. I think they only made up their mind about that when they cast Cumberbatch. Both reimagined Khan and reimagined Mitchell could have done everything in the film, secretly recruited, super strength, magic healing abilities, and all that.