I have a great deal of respect for the work Mr. Bradlee has done over his career, but you have a good point here.
I remember during the protests over the Summer when some reporters were attacked as part of a larger group doing nothing illegal, their response was not "we have a 1st amendment right to be here, but "hey, you can't do that. I'm with the press!"
That implicit privilege due to their position is perhaps more infuriating to me than even the accuracy of their reporting.
We see it in their version of the 1st amendment:
"Congress shall make no law abridging the freedom of speech of the press" FULL STOP
Meanwhile, they have no problem accusing someone like Julian Assange of "hiding behind the Constitution."