Late to this thread, but in case these weren't addressed -
From my personal experience - I've been interviewed for roughly a dozen national publications (including CNBC, LA Times, CBS, BI, CR - never NYT) for dozens of articles and only once (the Verge) has anyone ever called to fact-check.
Fact checking is usually done by consulting a different source to try to verify information you gave - either another person or a written source. This is not a fine-toothed review as a teacher might do on a term paper, and doesn't eliminate all inaccuracies but tries to minimize them.
When they're made aware of errors, they acknowledge some of them in a subsequent paper, usually things like names or dates. Today for the first time I see a correction in a crossword puzzle clue.
https://www.nytimes.com/section/corrections+1, every MSM publication has refused to let me see a pre-publication draft. Why was that done here?
I was also surprised by this, not only because they don't want you to demand change, but also because they don't want to be scooped on their scoop, as happened here. The pre-pub critiques said things like "They will probably say..." and elsewhere I read that they were sent an outline, so it's not clear how detailed that draft was. One tweet that included the message from the reporter said it was to give the recipient a chance to add info, can't remember where I saw that.