That really does nothing.
Not all dual citizens have two passports and those who do only use their American passport.
The country where I was born views me as a citizen and there is nothing I can do about that.
Having dual nationality does nothing for me, except that on the very slight chance that an extremely major calamity occurs in the United States, to the extent that society collapses, I have a place to go.
Many dual citizens have two passports. They use their US passport to travel from and to the US, but use their other passport for travel elsewhere. Having a passport from a EU country allows you to travel freely within the EU and you can work and stay for as long as you want.
You can denounce your citizenship in the other country similar to what Ted Cruz did with his birthright citizenship in Canada.