As someone who practices before both courts, I’ll take a crack at this one. The short answer is it was filed in the state court for exactly the opposite reason.
The case was filed in the Supreme Court of the State of New York, New York County [Manhattan]. (For historical reasons, the Supreme Court is the trial court of general jurisdiction in New York, equivalent to the Superior Court in most other states. Our highest court is the Court of Appeals.) New York County Supreme is actually a quite well regarded court, particularly its Commercial Division, though this action would likely not qualify as a commercial case. However, under the New York procedural rules and the case law interpreting them it can be easier to get a premiminary injunction in New York state court than in federal court.
In addition, the state court judges are likely less familiar with the First Amendment precedent that would protect Franken than their federal colleauges.
From a less purely legal perspective, just about any federal judge sitting in the Southern District of New York would have reacted more or less like Judge Chin did (though likely with less humor), calling Fox’s suit, if not utterly frivolous, at least deficient. On the other hand, there is the possiblilty that the Fox suit could have gotten some traction before some of the judges in the New York state courts.