I think autonomous support agents are just missing the point. LLMs are tools that empower the user. A support agent is very often in a somewhat adversarial position to the customer. You don't want to empower your adversary.

LLMs supporting an actual human customer service agent are fine and useful.

How do you prevent your adversary prompt-injecting your LLM when they communicate with it? Or if you prevent any such communication, how can the LLM be useful?