There are legitimate support cases that could be made better with AI but just getting to them is honestly harder than I thought when I was first exposed. It will be a while.
With "legacy industries" in particular, their websites are usually so busted with short session timeouts/etc that it's worth spending a few minutes on hold to get somebody else to do it.
These people don't want the thing done, they want to talk to someone on the phone. The monthly payment is an excuse to do so. I know, we did the customer research on it.
Again, this is something my firm studied. Not UX "interviews," actual behavioral studies with observation, different interventions, etc. When you're operating at utility scale there are a non-negligible number of customers who will do more work to talk to a human than to accomplish the task. It isn't about work, ease of use, or anything else - they legitimately just want to talk.
There are also some customers who will do whatever they can to avoid talking to a human, but that's a different problem than we're talking about.
But this is a digression from my main point. Most of the "easy things" AI can do for customer support are things that are already easily solved in other places, people (like you) are choosing not to use those solutions, and adding AI doesn't reduce the number of calls that make it to your customer service team, even when it is an objectively better experience that "does the work."
We've found that just a "Hey, how can I help?" will get many of these customers to dump every problem they've ever had on you, and if you can make turn two actually productive, then the odds of someone dropping out of the interaction is low.
The difference between "I need to cancel my subscription!" leading to "I can help with that! To find your subscription, what's your phone number?" or "The XYZ subscription you started last year?" is huge.