Unicode involves more than the set of glyphs and their encodings; it also involves properties, etc. However, it can be an attack vector even ignoring that stuff; it does not have to be Turing-complete to be an attack vector. But, the specific kind of attacks depends on the application.
Different kind of character sets and character encodings will be good for different purposes. Unicode is "equally bad" for many uses.
Yes, Unicode is too complicated and too messy, whether or not it is Turing-complete (it is complicated enough that maybe it is Turing-complete; I don't know).