I have Global Entry and Global Entry card. Even though it says my US citizenship on it, I wouldn't rely on it as solid proof. It may be accepted by CBP, but acceptance by other agencies is to be proven, there's not much data on whether SSA would accept it to prove US citizenship. Federal employers, I can almost guarantee, wouldn't see Global Entry card as credible proof of citizenship. I'm saying this, because I know somebody who's US passport and US passport card weren't sufficient to prove citizenship. Employer demanded naturalization certificate specifically.
Global entry is also available to citizens of certain countries like Germany, without even having any permanent status in the US. A German citizen can get GE and travel to US on ESTA as a tourist.