Did the ALMC deliver letters to every single address in the United States? I don't think they did. They weren't trying to "compete" with the USPS so much as skim off the profitable routes.
The USPS does not deliver to every address in the US.
I don’t know what specifically causes the USPS to deliver to a specific area, but there are entire incorporated cities, some with reasonably large population and density, that don’t receive mail at all except to the local post office. Even UPS and FedEx’s low-cost all-but-the-last-mile services regularly lose packages because they hand them off to the USPS.
They were not even trying to do that, Lysander formed the company with the expressed purpose that the government would sue him. he wanted to get the monopoly protection law for the USPS ruled unconstitutional
I don’t know what specifically causes the USPS to deliver to a specific area, but there are entire incorporated cities, some with reasonably large population and density, that don’t receive mail at all except to the local post office. Even UPS and FedEx’s low-cost all-but-the-last-mile services regularly lose packages because they hand them off to the USPS.