That's the point: If the barber has a lot of younger customers, the reason that most of the bookings happen on the phone or via text message is that the barber doesn't have a good website set up with a decent booking system that the customers could use instead of the phone.

What OP could have told the barber is this: "We'll put up a system that enables your customers to schedule a time automatically - that way you'll only have to schedule times manually in the 20% of cases or so when a customer prefers to call rather than use the website".

It's not rocket science - plenty of barbers have such a system.