Nobody really knows what is meant by "officer of the government." It might include the president. It might not. A president occupies a unique position under the Constitution.
Basically what the judge did was punt the issue to the appellate court.
The judge found that Trump incited an insurrection (a finding of fact) but didn't find that he was an officer of the federal government (a matter of law).
If an issue that goes to an appellate court is an issue of law, the appellate court reviews de novo, which means that no deference is given to the lower court.
Welcome to the world of constitutional and statutory interpretation.
Basically, the Constitution means whatever the Supreme Court says it means because the Supreme Court is the final arbiter and the Constitution is stuffed full of phrases and words that can be interpreted in different ways.
An interpretation of the word "reasonable" in the Fourth Amendment has literally filled books.
The Third Section of the Fourteenth Amendment was added to the Constitution after the Civil War to keep former Confederates out of the government.
The idea was to prevent backsliding.
It didn't work because, by the late 1890s, the government and Supreme Court were stuffed full of Confederate sympathizers who rolled back the advances made during Reconstruction and gave us racial segregation.
You can keep out the insurrectionists but not the insurrection sympathizers.
I've never practiced in New York, but I think it's safe to say this is a slam dunk in the Loser Department.
Complaint #1: The court, in newsletters, linked to articles that the Trumps think are unflattering to them. (I have no idea what this is about. It's pages 3-4)
. . . and is appropriate only whenever circumstances arise that cast substantial doubt upon the fairness or impartiality of the trial.
Because the underlying premise is wrong (she was a co-judge) I'm sure that a law clerk's personal business doesn't effect the fairness of the trial.
The only thing that might have created the circumstances that cast "substantial doubt" are all Team Trump's attacks on the clerk, and in my world (not NY) that can't be a basis for a mistrial.
Whether this motion violates the gag order is an interesting question.
I didn't follow the trial as closely as others, but it was clear that the lawyers for Trump planned to raise the issue of the clerk on appeal. They made a point of creating a record, so the judge knew they would do this.
I'm not sure a gag order can prevent a person from raising issues in motions and on appeal. (Possibly nobody knows because I doubt anything like this ever happened.)