I think the primary benefit of having your own driver is that you get to put your branding on the car and the driver's uniform. It's not clear to me how much seeing that little "Pizza Hut" car topper impacts sales.
Restaurants can also make use of drivers they directly employ durning downtime to do other stuff like sweep up.
https://www.ups.com/us/en/services/knowledge-center/article....
Believe me I understand why it seems possible, but this is one of those cases where real world data doesn’t fit abstract models very well. One example is during peak times both the restaurant and it’s drivers get overwhelmed so preparation time is increasing. Another example is if their returning to the same location they can easily return empty insulated boxes, but that’s problematic if their doing pickup from 10+ restaurants a shift.
On the other hand someone showing up at potentially hundreds of restaurants is hardly going to know each of their systems very well or be recognized on sight.