How to Convince a Business they Need a Website

In short, don’t.
There are far more opportunitites out there for building websites than there are people that can make them. And with how quickly technology changes and sites become “outdated”, I think that trend will be around for a long time.
There are new SEO practices, new programming languages, new design styles, and other trends (Progressive Web Apps!). There are tons of businesses out there who desperately need help with what they have, or are trying to find out what they don’t have. If you are having to convince people that they need a website, you are looking for the wrong thing.
Look for problems, and solve them for money. That is what business is.