I've worked as pizza delivery. The tipping and per-delivery payments to drivers is why it works.
Roughly speaking, a delivery driver in my area nearly a decade ago earned roughly $20/hr after gas and car expenses, half in hourly wages @ state minimum and the other half in tips + $1/delivery payment. They also took about 3 deliveries an hour when fully utilized, which wasn't for 100% of their shift. The wait time got used productively, combination of answering phones, making pizzas, or at worst folding boxes.
This combination means that you're employing people at a significnatly-above-minimum-wage level, but their excess time waiting for deliveries gets paid at minimum wage and used to do customer service. And in the worst case when things are so absolutely slammed that there is no downtime, well, that's when you have to let phones ring (or answer briefly with "thank you for calling, our wait time is currently two hours, would you like to place an order?"). Plus these drivers know the store and the area well and from that alone should be significantly more competent at resolving issues - often enough someone calling in about their order will end up talking to the person who just tried and failed to make the delivery.
With Postmates and the like, though, the time spent waiting for a delivery has no useful work to fill the time. It's also not paid, but the gig has to justify itself economically, which means that drivers have to make up for it through increased per-delivery payments.
Basically, the difference is that store-employed drivers can do useful work while waiting to engage in deliveries, while app drivers can't. This is a very significant efficiency gain.