They are correct. The USPS has a universal service obligation. If you send a letter or parcel to any address in the US, the post office must deliver it.
This means it is an org mandated to accept your request for service at a reasonable price regardless of your location of residence. There is an important distinction between that, and being mandated to receive mail service. That is not a stipulation of being alive in USA.
[0] - https://www.uspsoig.gov/blog/postal-service-and-its-obligati...
They don't have to. Just have mail sent to:
Joe/Jane Drifter
General Delivery
TownNearWhereYouSquat, <State> <ZipCode of post office>
Or at least that's how it's been done for decades.