I feel delivery companies should bid for providing service for the last couple of kilometers, then the winner should get an exclusive contract for some time frame. That's how it's done for other services like regional rail public transport.
A forced bidding system seems to give gameability and artifical barriers to entry for little gain. To give an anchhacic example of the sort of problems it would be akin to requiring a business that can serve all of New York City competively before bidding. Having a mail room in the lobby of a large apartment would violate that due to serving only residents.
If there was just a single company responsible for some "last mile" area, smaller vehicles like this electric model [1] would suffice. They would distribute parcels from a depot that is filled by larger long distance trucks that do not need to belong to the same company.
The last point is important: If DHL, FedEx, UPS etc. each have their own final storage buildings and delivery fleet, those depots could not all be situated close enough to the target addresses.
Ask any Brit how privatized rail's working out for them.