To their credit, they refunded me when I complained about this.
Personally I think it should be illegal to renew annual subscriptions without some kind of reminder that the bill is coming up, or an opportunity to cancel at some point, especially if you never even used the service. Heck, I might go so far as to say that annual billing should not be allowed.
"Your payment method will be automatically charged $4.00 every 4 weeks for the first year ($1.00 per week).
It will then be automatically charged $17.00 every 4 weeks thereafter, starting on (date) ($4.25 per week).
Your subscription will continue until you cancel. You can cancel anytime."
In your case, you may not have checked a box, but I'd be shocked if there wasn't similar language saying that you were opting in to a recurring payment. (It's not as if they're making you read a 5000-word EULA - it's right there front and center on the page in my case.) Not to mention that this is pretty standard practice across subscription-based businesses.
I do think it is good form for subscriptions to give customers advance notice of when their subscriptions will renew, though, and most don't. I would go so far as saying that I wouldn't mind if this was a requirement.
What really bothered me was they took the payment method itself as an indication that I wanted a different type of subscription. Check? 1 year subscription. Cash? 1 year subscription. Credit card? oh ho! You obviously want to subscribe from now until the end of eternity! And yes, they told me on the phone that my use of a CC was indication that I wanted to autorenew and that there's not a way to pay with CC that doesn't autorenew (at the time I spoke with them in early 2018).
Basically to the NYT: "because of the payment method you used, we are able to keep billing you" = "you must want us to keep billing you!" and that is complete bullshit.
May I ask how you subscribed? I agree it's pretty shady to offer multiple payment methods side-by-side with different terms, but I also haven't seen a lot of online merchants accepting cash/check.
I should have made it more clear that auto-renewing seems like the default specifically for subscriptions made online through the web or an app.
I had to call and pitch a fit to get them to at least stop sending papers (to my old address, I don't live in the USA any more) and give me a prorated refund. They were completely shady and untrustworthy, I'll never do business with them again.