This is correct. I know several people that have US + Italian, and I've been considering it myself, although Italy doesn't seem like much of an "escape pod" (per another comment) at the moment, the way things are going there.
What do you mean "doesn't recognize other citizenships"? Dual citizenship is legal.
You still have to bring your US passport to pass the customs. That is not the case for other counties that would recognize any valid passport.
Countries are free to define their own travel requirements for their citizens, which they do for security reasons. What you describe is a very obtuse definition of "does not recognize other passports."