I have not.
Balsamic- at least the standard dark kind (I haven't tried white balsamic)- has a slightly sweet taste. Having accidentally made a chicken soup with sweet onions instead of the normal kind, I can say that sweetness in chicken soup is best kept to a minimum. Usually, the carrots are sufficient.
A little lemon juice or citric acid, OTOH, is a winner. Gets you into the Greek/Tex-Mex flavorhood.
A little lime, mint or basil gets you into chicken pho territory, which is also good.