It says freedom of the press....like the 2nd Amendment, it doesn't give limits.
You can't use the "freedom of the press" to defame someone - there are reasonable limits.
I'm missing where anonymous sources are protected.
They're free to report. The law is also free to investigate crimes and elicit testimony from those of possession of knowledge pertaining to such. Nobody's stopping them from reporting what they know in full. It's that they won't that's the problem.
That kind of misses the point. The Second Amendment doesn't give you the right to gun down your neighbor just because you feel like it. You can own a gun, but there are limits to how you can legally use it. The same with the First.