Of all the services I've tried in this space, only Postmates lets you request items from businesses they aren't partnered with. This was a huge win once when I needed groceries delivered quickly.
They all do this now, to one degree or another. [1] [2]
The economics are even worse (because they don't get a cut from the restaurant), so these types of orders are either very high fee for the consumer, a loss leader, or both.
I've heard that on Postmates you can use the write-in functionality to order from a business that isn't on their platform at all (you order from a nearby business and in the write-in field tell the Postmate to go across the street, etc). I'm not sure if many of the other services provide a way to do that.
So they can go to the restaurant and order for you? That's interesting. I can't imagine it being very efficient or very fast though, how do they manage to not lose tens of minutes per order on non partnered restaurants/stores?
You can also order non-food items on Postmates (e.g. stuff from Home Depot).
Also anecdotally, the probability of an order actually arriving and being correct seems to be higher on Postmates. Perhaps the drivers there are more motivated to do a good job.