Well, I agree with number 1 but number 2 is well....I don't know what your logic is with that. Tuners do exactly what you said they don't do. They DO correct exit time with muzzle position, which is what tune is, tuner or not. So yes, they very much will put the gun in tune with the load...to the potential of the load in that rifle. Maybe I'm reading something wrong in your post. If so, I do apologize for that.
I do agree that bbls don't only vibrate vertically. Part of the vibrational oscillation does have a horizontal component. When not in tune, it shows on target clearly. Due to gravity, it is vertically biased and the groups show up as diagonal in shape. Pure horizontal is rare except when wind is the main factor or at least a factor along with tune. IME, there is NO tune condition that can vibrate ONLY horizontally though.
I thought this to be well hashed out and well documented so again, if I misread your post, I do apologize. Otherwise though, I'd be happy to share a lot of data learned doing vibration analysis testing along with years of experience using them to help you understand better what and how tuners do what they do. I don't mean to be condescending at all with this statement but if I read your post as you intend, it's way off base.