Couldn’t the governor let her loose via an EO?
No. The contempt power belongs to the court alone. No statute involved, so there is no criminal violation. Contempt is a prerogative of the court, and is entirely in its control.
If the court abuses this power, it undermines its legitimacy both according to its own rules, and also in the eyes of the public.
Complicating the situation in this case is that the offended judge is a federal judge, outside of the reach of the state. I assume the clerk is incarcerated in a federal facility. If she is incarcerated in a state facility, or if the judge orders the state to incarcerate her, then yes, the governor can tell the feds to stuff it.