When you pay your medical bills it’s still an explicit payment.
Businesses probably need contracts in order to function, but they are overused in business-to-consumer transactions. That's the underlying problem that we should solve -- you should be able to walk away, no questions asked, from paying for a newspaper or magazine.
Attacking the latter might make a large difference even if the former remains unsolved. The New York Times can get away with making cancellation difficult because they have the power to unilaterally take money from their (former?) customers. But, if anyone could trivially revoke their authorization to charge them money, I doubt that the New York Times would actually try to sue or collect from their customers en masse. Sure, they could try, but that would be a fantastic way to piss everyone off and to recover very little money.
It would be worthwhile to consider not letting "click agree" create a binding contract. I think I'm in favor of that.
I agree that things like newspapers don't need to be a subscription or have a contract. On the first of the month they should just pop up a dialog that asks if you still want the subscription, and if so, it charges your card for 1 month. I would certainly like that, but it does carry a risk on my end -- if they go out of business on the second of the month, I'm stuck paying for 29 days of the subscription I can't use.
Like I said, the big problem is not being able to cancel. That's why I buy subscriptions through Apple -- there's always a cancel button. I think we should make that mandatory for every subscription provider.