The big advantage for online ordering is that it removes the time pressure of being on a phone call with a busy live human being. This not only saves the store labor costs, but also increases sales - the time pressure defaults "do I want X" decisions to "no", and given the laxer pace of online ordering, people tend to spend more.
And I would think it prevents mistakes in ordering. Via the phone (a pizza shop etc is not generally very quiet) I often got the wrong order (but not completely wrong; chicken instead of pepperoni, not pasta instead of pizza) and/or delivered to the wrong address etc.
A lot of chains did centralized call-taking and modems to send the order right to a triplicate dot matrix printer fairly early on. With a fax backup I guess.
It also helps HQ stop under-the-table sales, or blocking the phone lines when they’re busy.
I think a place I use still has its orders print on that.
