It sounds like there'd be some risk involved..
Or host it in a country that likes to irritate the US government. Iran comes to mind. Venezuela would surely be open to it as well.
If Osama Bin Laden can find hosting there is no reason why a group of clever engineers can't as well.
This is the internet, after all.
What would happen is, they will call CIA first to verify the authenticity. Then CIA will threaten them and will claim this is 'classified' and if they don't immediately turn it over, heads will roll, hard drives will be shredded, offices will be closed for an indefinite time for investigation and so on. NYT will not touch this stuff with a ten foot pole.
From a more general perspective. Large news media outlets have very close and 'cosy' relationship with the government. They treasure that relationship as that gets them quick and timely access to government related news releases. That is just news that writes itself. Whatever spews out the Department Of State goes straight to the viewers/reader. All news media outlets need to do is add advertisement on top. Being cut off from that, means spending serious time and money on investigative journalism. Flying people around the world to dangerous areas, spending months on finding good leads and researching. NYT will just not take that risk.
"Bush Lets U.S. Spy on Callers Without Courts" http://www.nytimes.com/2005/12/16/politics/16program.html
This is most likely about murdering civilians in a botched air strike by CIA. CIA can and will shut a place down if they believe there is classified information there. I have heard of offices being raided and all hard drives shredded.
I think your belief that big, supposedly trust-worthy newspapers like the NYT are independent and publish such things, is not warranted anymore. They only publish what suits their interests, i.e. in some specific cases they may refrain from publishing, but that is exactly what counts, sadly.