Ended up on a 2h call with Chase which resulted in building Relaya to automate this. For simple tasks (e.g. checking if something is in store, making a reservation that isn't on opentable, etc.), it will just do the full task. No need to dial anything.
For more complex tasks e.g. applying Chase Sapphire credits to rebook an existing flight, it will wade through all the menus and holds and connect you with the agent directly.
At the very least, only talk to humans and reduces an avg. 20-30 min call to 2-3 mins (~20+ ppl have tried so far).
Next, I imagine will be able to automate a certain % of the more complex calls as well. Looking for feedback.