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.
If voice is crucial, save yourself answering the phone with some sort of Twilio setup to handle the majority of calls with a series of questions about timing and duration of appointment. In the preamble mention online bookings and see if the majority stick with phone.
I know I'd vastly prefer a web booking system for almost everything - restaurants, car service, medical appointments, etc.
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.