That's what companies fail to consider when they deploy these "retention strategies." Every roadblock you put between me and cancelling is ALSO a roadblock between me and resubscription in the future.
I've subscribed and cancelled Hulu+ three or four times when there was a lull in the content. But I'd have no hesitation in signing up again because I remember how hassle free it was to cancel.
Contrast that against LogMeIn which has the same strategy as NYT by the sounds of it (call to cancel, takes 15+ minutes). When I needed LogMeIn for a new project, I looked around for alternatives instead because I remembered how big of a hassle it was to cancel. Ultimately LogMeIn didn't get my repeat business, not due to the financial cost, but the hassle cost of them.
Plus of course word of mouth like this is hugely damaging for these strategies. I won't be getting a NYT subscription now.