Judge Carter could not make this finding because what you have described is the definition of “quo warranto”. Under separation of powers, quo warranto is reserved to Congress which has delegated its quo warranto authority to only the DC Circuit, not to any other federal circuit court, such as Judge Carter's.
Taitz and Kreep even admitted this, but unrealistically (or disingenuously) thought they could persuade Carter that he should hear a quo warranto claim because in their view, the DC Circuit had shown bias. I'm not a lawyer, but I don't think any federal judge would grant jurisdiction based on such a vague, unsubstantiated allegation by a plaintiff against a fellow sitting federal judge.
Now, with D’Onofrio’s impending quo warranto filing in the DC Circuit for the Chrysler dealers, we have potentially the right plaintifs in the right court with an available remedy, if SCOTUS will agree ultimately, that quo warranto applies to POTUS.>
Quo warranto is an alternative legal path. It is not the only path.