What you're saying is really interesting. If race is a social construct, then it can be overcome independent of skin colour. As you said, Italians, Poles, Slavs "became" white over time, and Jews were enjoying "White privilege" by serving in congress. By that measure, a Black person who transcends prejudice, who achieves high status or acceptance, serving in Congress, presiding as a judge, attending as a doctor, teaching at Yale, also becomes White. Isn't that a source of friction, trying to identify as Black while society sees you as no different than Whites?
On Whoopi, what she said was racist. Though she might not be able to distinguish Jews from Germans, Europeans believed at the time that Jews could be distinguished by appearance. That's not ethnicity or religion but race. Isn't that precisely what you're saying about Black Americans? To defend her is no different from saying "All ____ people look the same." Why should being Black make that okay?