“When the president of the United States asks for time to make an address to the American public, I don’t think you can say no,” Ted Koppel, the longtime ABC anchorman, said in an interview. Any concern about misleading claims, he added, “is a judgment you can only make after you hear what he has to say.”

Others argued that traditional ways of covering the president should not apply to the norm-breaking Mr. Trump. Former aides to President Barack Obama pointed out that his request for airtime in 2014 to discuss immigration reform was turned down by the networks, which called the subject overly political. One aide, Dan Pfeiffer, wrote on Twitter after Mr. Trump’s remarks on Tuesday night that “the Networks got played.”

Still, producers at several networks cited the uncertainty surrounding the shutdown, and an impending suspension of federal paychecks, as inherently newsworthy.

Executives were juggling a slew of concerns, some tied to public responsibility and others crassly commercial. Tuesday is a relatively slow night for prime time, so airing the president’s address would not pre-empt a ratings magnet like a National Football League game. Executives in New York are mindful of their national affiliates, many of which cater to red-state audiences that might be antagonized to discover their local station is declining to show Mr. Trump’s remarks.

Even as the news landscape has fractured, the four major networks still have the biggest audience in mass media. Some journalists had questioned if a different approach was necessary.

“There are so many sources of news now, but the networks’ policies haven’t changed very much,” Andrew Heyward, a former president of CBS News, said before the speech. “One very attractive option would be to promote the fact that it’s live with a crawl, and then if it turns out to be newsworthy, you can do a special report. You then pull selected sound from it and give it context, as opposed to having to get trapped in something and having to react to it.”

For those who sought counterprogramming on Tuesday, one former Trump associate offered an alternative.