Not sure how the Apple Card system works, but Visa has a program that I was affected by. I had a card that needed to be replaced before its expiration date. I intentionally did not go online to update it anywhere. I wanted to start receiving emails from all of the sites that the previous card was registered as I was specifically looking for sites the card might have been registered without my knowledge. However, Netflix pays into the Visa program so that they can find out that new card number and automatically start using it without me doing anything.
I don't know if the NYT does this, but with some services, if you invalidate your card without formally canceling, they treat it as ever-increasing unpaid debt and eventually send you to collections, and then you've entered the hell world of aggressive debt collectors and potential damage to your credit.
Surely changing a card number doesn't communicate your intent to unsubscribe from any service where you have an ongoing subscription. It prevents the bill payment, but doesn't stop the subscription and any costs associated.
There used to be a service which allowed you to create virtual CC numbers. That way you could create a new number for each service. Just like some people create a new email address when they sign up online.
Capital One and Citi let you create virtual CC numbers.
Bank of America used to, but they discontinued that feature a few months ago, saying that with electronic wallets such as Apple Pay and similar that do not disclose your card number to the merchant there is no longer a need for virtual CC numbers.
I've got a couple Capital One cards and their virtual numbers are quite convenient. They are created from their Eno browser extension. Invoke Eno on the checkout page of a site, and it lets you pick and existing virtual card or lets you create a new one for that site.
I use it on Firefox, but recently the Eno extension disappeared from the Firefox extensions store, and they no longer list Firefox as a supported browser (although existing installations continue to work). If this is permanent it will not be as convenient for Firefox users--they will have to use Chrome to create new virtual cards, which they can of course still use in Firefox.
On your account pages at C1 you can view your virtual numbers, suspend them, and delete them but you cannot create new ones. That has to be done through the Eno browser extension.
I have no idea what Citi's virtual card system is like.
Privacy.com still exists and does exactly that. Most services that offer free trials (like the $300 Google Cloud Platform credit/trial) don't accept virtual credit card numbers because it would allow someone to use an infinite free trial.