Navigating the "press 1 for billing" menus on a smartphone does involve prodding on a screen. 😀
I don't think there's anything more human and humane about "press 1 for billing" than selecting options on a screen. The "press 1 for billing" things often have multiple levels to negotiate before you get to talk to a human. Selecting options on a screen seems less annoying because they're all there for you to look at and choose from in your own time, you don't have to listen whilst someone reads out a load of options that aren't what you want, or think "was it actually the option before last I wanted?" or "great, my two year old shouted and I missed what that option was" or "how do I make it tell me the options again without waiting for it to eventually finish reading them all out?". But I agree a set up which involves you dialling a number, then selecting from options that appear on a screen would fail. And it wouldn't be usable by a huge number of people anyway.
An app which lets you navigate a visual equivalent of "press 1 for billing" and then initiates a phone call that connects you to the relevant department might be somewhat successful. But simply publishing a list of phone numbers for each department on the website would work for a lot more people and be vastly cheaper.