If you try to compete with DoorDash using that model, you will either have to charge much higher prices than they do, or you will lose money on every transaction. Either way you will go out of business as long as DoorDash or similar companies are able to compete with you.
Well, if the other employer existed first, you might be right. The other employer might be able to deplete the labor pool of unemployed people with cars to a sufficiently size that DoorDash wouldn't be able to get started, if there were so many people willing to pay for grocery delivery (at the higher prices the other employer charged) that they could employ all or essentially all of the potential deliverypeople. But if there were more potential deliverypeople than potential customers, including at the lower price point that DoorDash can (hic) deliver, or if DoorDash already existed and could therefore prevent the new company from breaking into this two-sided market, then no.
The reason the sequence matters is that, for DoorDash to need to raise its payments to its workers in order to keep them from running to the other employer, that other employer needs to be able to hire all of them, or at least so many that DoorDash finds its access to the labor market significantly restricted. But clearly a new entrant starting small will not be able to do that; they can only do it once they get big. But they will not be able to get big if they are charging higher prices for the same service as DoorDash.
So, no, much more reasonable assumptions are available to justify my inference.
These companies have discovered that there is a large market of people whose need for quick cash with no friction exists and is willing to lose money (in terms of car wear and tear) to do so.