Bob I can easily destroy your perfect theory by saying "у меня есть глаза"
(Folks, don't dare to say "у меня глаза", it sounds like a disease, like I have a headache, or I have a corn on my toe)

However, everything you said about hair is right. Maybe it's all about the adjective, like "У меня зелёные глаза", or about the possibility of rewording like "I have green eyes -> My eyes are green" while you can't say "my headache is in the head"