The information that so-and-so parties provided some information (without disclosure of that information) in response to a lawful request will usually not predudice a trial.
What gag orders are for is a) avoiding tipping off the subject of an active investigation b) avoiding general knowledge or disclosure of key sources of information and investigative methods used by law enforcement and c) concealing the general scale, nature and purpose of surveillance activities from the general public.
I agree, but the only time I think it's justified is when it's to protect the right to a fair trial.
I don't know if humans are able to ignore evidence they have heard outside of court. We are not good at only including one set of information when making judgements.
I agree that free speech has significant value and restrictions should only be put in place when required to uphold other rights.
I'm sure you're not alone in your opinion that a jury is able to ignore what they've heard about a case in the media. However you do disagree with the current judicial system of most developed nations. That doesn't necessarily make you wrong. I think there is merit to the argument you're making, I just am not convinced that people can ignore information like that.
https://juryanalyst.com/blog/the-power-of-media-coverage-how...
https://www.canlii.org/en/commentary/doc/2019CanLIIDocs2798#...