From a legal POV they don’t. From the POV of reaching a huge, mostly elderly (i.e. voting) audience that gets its news exclusively from Fox, they really do. Leaving out that audience of 10s of millions, which isn’t going to switch because of a White House decision, is self-defeating even taking into account its dim-witted nature. The question is more how to handle the bad actor you’re stuck letting in the room, and Psaki does this masterfully.
That was the real failing. The other outlets could have taken that opportunity to say exactly what Fox News is (i.e. not news). Instead they fretted about (admittedly likely) retaliation from a future Republican administration and went into their usual (Fox inspired) BS cant about fairness and balance.