Some sort of myth seems to have been created that news reporting can't use anonymous sources. It's not only wrong, but not disclosing sources is an absolutely foundational accepted element of journalism.
But if you're skeptical about whether they completely made it up, published a rumour without any confirmation or are deliberately substantially altering it in how they portray it - that's an extremely serious allegation for a reputable news organisation. For example, I think that would stretch beyond "reasonable" in most cases for NYT based on my observation of their practices over time.
There is no way to tell if a lie originated with two anonymous sources or with a reporter with an agenda. The "trust but verify" approach only works if something can be verified. IMO the nyt damages its reputation when it posts rumors which cannot be verified.
This is just a more general case of having a trusted broker. How do you trust your bank? How do you trust airlines to be safe? How do you know your doctor is competent? You don't demand first hand evidence for all of these. You trust a regulator to oversee them and the regulator may absolutely rely on evidence that is not made publicly available. A lot of society ceases to operate if you throw out reputational trust.
Airline crashes are regularly in the news when they occur and are easily verifiable.
My doctor has a licensing board and such to report to.
What recourse do I have if the new york times posts something not truthful that can't be verified? How would I even know? I am not the only person to not trust the new york times, they do not have near the level of public support as the Medical/Airline industries. Given the fundamental lack of accountability inherent to anonymously sourced articles, why wouldn't they be abused?