So call their bluff. Unless you're dealing with the support for some small/fly-by-night company, phone support is going to have someone overseeing them. If they absolutely refuse to connect you, hang up and get a new agent and ask to speak to the supervisor from there. Make sure to provide the supervisor with the calling agents details.
All of them have a supervisor or superiors of some sort. But in some companies they do not touch the phone, they overseer the process. I can tell you from personal experience that if the customer support doesn't solve the problem then there isn't a next "level" of support called the supervisor, but only more formal options to communicate remain; think complaint department, mail, lawyers.
I dealt with this BS during a brief stint with Straight Talk Wireless when I was between regular carriers. I was trying to port my number from Straight Talk to Ting (a wonderful Sprint MVNO) and my account ended up in limbo. Every time I called, I asked to be escalated to the next level of support, and every time I was told that there wasn't a next level, that Tier 1 was it. I pushed it every time, and ended up with a "supervisor" who also read from the prompts, but in a slightly more understandable accent.
I finally got Supervisor Number Three to understand what was going on, and he gave me an entirely different phone number to call, which ended up being their corporate office in India. After talking to three different people there (again, "escalating" laterally) I ended up with someone who claimed to be the director of support, and she was able to unlock my account and allow the port to complete. She said that she had never seen a situation like that before, and assured me that if I ever came back to Straight Talk I wouldn't see this issue again, as they would "fix the porting process" to enable their Tier 1 staff to push the port through.
This entire process took three days of calling from my wife's phone, since mine was in limbo and unusable on either service, and probably shaved several months off my lifespan due to the stress and anxiety over the whole thing. One thing is for sure, I will never, ever use Straight Talk again, I don't care if someone gives me a $600 Straight Talk-compatible phone.
In comparison, on Ting, I call their support line and within three rings I have a real person who has full access to my account and can do pretty much anything I need to with it. Just out of curiosity, I asked a Ting rep how the cancellation process goes if I wanted to go to another carrier, and he said they would help me make the transition as smooth as possible, and would hate to see me go but he wouldn't want to hold me back if another carrier offered better coverage or rates.
Comcast, Straight Talk, and other service companies could learn a lot from them.
If we forget for a second the completely incorrect behavior of the Comcast operator.
A correct response to a unsatisfied customer would be: My supervisor doesn't talk to customers. I am the support for the company, if you are not satisfied with me, you can talk to a different operator or file a formal complaint.
This is an actual correct response to a conversation that has gone awry,( of course again, assuming the operator is behaving like a decent human being ).