I've cancelled by NYTimes subscription just a few months ago. Didn't have to do any phone calls-- just answered a few questions in the chat, right there on their website.
They "generously" let you use an IM chat now, yes. But there's no excuse for it not being a simple button. The only reason is so they can raise the effort barrier just a bit more.
Had a similar experience when I left my previous ISP. I didn’t even have a choice to keep the service as I was moving. Upgrading the service was an easy click, but cancelling consisted of reading through their help pages to find a number to call, which of course was only open a few hours per week. It explicitly said that this was the only way to cancel.
Instead I simply sent them an email insisting that I wanted to cancel, and that I refused to talk to them on the phone to do it. They replied with a “great offer” to reduce the price. In my reply I simply insisted on cancelling. I didn’t hear from them again, except for a confirmation by post a while later that the service had been cancelled.
Of course, this is much more effort than it should be, especially an emotional effort because being treated like this is very upsetting. But it might be worth trying.
They wouldn't let me cancel until I answered the question "how are you doing today?" I said "I want to cancel" but they just repeated the question until I finally said "fine" and then the conversation could proceed. Fun times.
