Criminal courts have jurisdiction over individuals when they are charged with an offence, either after indictment by a grand jury or whatever other process is set out by law. They also have jurisdiction when a person is subpoenaed. When a person is subject to the court’s jurisdiction by either of those processes, they may potentially be subject to contempt proceedings, for disobeying the court’s process, disrupting, and so on.
There is also a type of contempt that goes by various names, for actions committed outside of court. I have no idea how that works in the US and in the different states.
However, when a person is charged with contempt, the onus, as with other criminal counts, is on the state to prove it.
What the criminal courts don’t have is a power to call people in, put them under oath, and interrogate them. There is a historical precedent for that type of proceeding, called the “oath ex officio” or the great oath. It was used by the ecclesiastical courts in England, and by the Court of Star Chamber. The abuse of the oath ex officio was one of the contributing factors in the political crises that led to the English Civil Wars. The abuse of the oath led to the abolition of Star Chamber in 1640, and the abolition of the Court of High Commission (also called the Court of Commissioners for Ecclesiastical Causes), originally in 1641, and then finally in the Bill of Rights, 1688, after James II revived it.
The abuse of the oath ex officio was one of the contributing factors to the development of the legal and constitutional right to silence, which Ms Habba could invoke if called to court.