What is the technology to stop them?
But they'll never take that leap of faith absent a law mandating it. There's not enough competition between the major issuing networks.
Right now, they can't, because they have no privileged way to cancel a service on the user's behalf. They can withhold payment, yes, but under current consumer contract law withholding payment doesn't imply cancellation. We should change the law so that it does. Once the law changes, they would implement "cancel buttons" and "subscription management" so fast your head would spin -- and then they would flood TV and youtube with ads about it, and they would bug you every time you logged into your CC statement for the next year.
... the way to do it would be to disincentivize subscription companies from having credit card companies involved in cancellations.
(1) Pass laws to empower credit card companies to cancel services on a user's behalf, if the user so requests + (2) Allow CC companies to collect a fee from the subscription service when they do so.
CC companies will implement the feature as a profit source, but subscription services will also be incentivized to flow cancellations through themselves (because it's cheaper) thus leading to more streamlined cancellation.