At least in France we passed a law that says that if you subscribed to a service online, you can unsubscribed online too (no idea if we're first, or last).
But, what do you mean by “take action on the entire customer account?” “Trick or bribe a human” is not a very difficult task, so it must be the case that there are some things a customer service agent can’t do.
Granting access, clearly a no-go for any service that might hold important private documents. But also, canceling an account on that sort of service could be pretty catastrophic… which means they can’t resolve problems in a direction that stops them from billing you…
For a while some sites only let you unsubscribe online if your address was in California, so people would change their address and then unsubscribe online.