That strikes me as unreasonable. We don't impose that kind of a requirement on package delivery on one's porch...why should we impose that requirement on paper mail delivery in one's mailbox? In no communication enterprise do we ever require 100% coverage. Markets finds some equilibrium with global, national, and local players coexisting. The same can be true for state-run enterprises.
And we absolutely do place similar requirements on new entrants in monopoly enterprises. I'm thinking about things like 911 service on cell phones/VOIP phones. Basically, we all need the "new mail" to work pretty much like the old mail it's replacing. It's in nobody's interest for an inferior system to eat all the profit and drive the existing system into collapse.
For example, if they want access to consumer mailboxes in an area, they need to cover 100% of the addresses USPS covers without routing packages through USPS. That way, they don't have the ability to surgically cherry-pick the most lucrative aspects and leave USPS with the crappy bits due to its mandates.
Given that USPS and FedEx are roughly $10 for a 3-day letter, I don't know which carriers would even want to deliver paper mail.