I wouldn't put it past they NYT to do it. They could say that for privacy reasons the billing department can't view the last time you were logged into your account, and boom, they can argue that if they didn't pursue payment, you could be getting features from your paid account for free.
Most recurring billing APIs provide some sort of notification or query about the status. They could reasonably do "for each payment we got yesterday, extend the expiration date of the associated account."
Sure. They could also let you easily cancel online. But they don't want to. And if you refuse to pay for a subscription contract you entered, but didn't cancel, you owe them.
My experience is internet companies don't collect debt the way you describe. It's the offline companies that do (ISPs, gyms, utilities, banks, etc)
Edit: they also need to inform you before sending your debt for collections, giving you time to get it right with them before any credit score issues. Seriously, people are so worried about keeping their promises to companies that don't reciprocate that concern. Most companies just deal with the consequences of breaking the rules. Why shouldn't we apply the same to them as their users?
From a legal perspective, would it be enough to send them an e-mail telling them that you no longer want to subscribe to the services before pulling the plug?
It really doesn’t. You wouldn’t even need the email. If this ever went to court the NYT wouldn’t win. Judges are human beings, not computers, and they consider circumstances. All you have to do is demonstrate intent, and not use the service so they can’t argue you received benefit without making an attempt to cancel.
Does PayPal allow that? If they are smart, they'd put into their ToS that in order to use PayPals subscription feature to receive money, you have to agree to let people cancel through it and honor this.
