Recently, there was much consternation about Judge Kacsmaryk's decision to delay posting notice of a hearing. Indeed, a coalition of media organizations actually argued that this decision could violate the First Amendment! Ultimately, the proceeding went as planned, with no disruptions or incidents. There were protests outside. And the event was widely covered by the press. Thankfully, democracy did not die in the darkness.
Throughout this entire process, I chuckled. People who had zero experience with federal district court litigation suddenly became experts. In reality, trial judges have vast discretion over their dockets and courtrooms. In any normal case, this sort of request would never have raised an eyebrow. And the information would have never leaked to the press. But, with the abortion ad-hoc nullification machine at maximum power, all the usual rules are ignored.
If you'd like some evidence of how much power judges have to keep their proceedings secret, consider the sidebar conference. Generally, everything a judge says is in open court. But the judge can ask the parties to "approach" the bench, at which point the judge and attorneys can have a private conversation that the witness, jury, and other parties cannot hear. Some courts have noice-cancelling devices that make it impossible to even hear anything. (The district court that I clerked in did not have that technology, and was very small, so the parties were asked to speak low, but not too low so that the court reporter could not hear them.) Generally, the court reporter transcribes these proceedings. But sidebars may be redacted from the public transcripts.