Does it say that the personality is not part of the body?
No, it just repeats the initial assumption. Your thought experiment already assumed that personality is not part of the body. The thought experiment's assumption that it is possible to transfer a personality to another body logically implies that the personality is not a part of the body.
It is much more realist and feasible to imagine that a mad scientist transfers your brain into another body. Presumably, all your memories, beliefs, ambitions, etc. would follow your brain. You would feel that you have the broadly same personality but not the same body. Yet, having a different body will probably change your personality. First, your memories starting from the day of transfer will be those associated with your new body, I guess that should count as a change in your personality. You may become the happy owner of a large fortune instead of being destitute living on the street. That will change your personality. People may turn out to be nicer to you once you get your new body and this may help you be happier and therefore change your personality. You may change your beliefs, your ambitions, etc.
If so, then personality is not only part of the brain but also a part of the body.
In this case, however, your soul would have stayed with your original body because God and only God, and possibly Satan, can change where your soul is, but you said 'mad scientist', not God or Satan, so the soul stays where it is.
Well, I guess this may be a problem.