I wonder if UPS, FedEx or someone else will take their place?
IIRC, UPS and FedEx already fly a lot of the Post Offices mail.
And the semis and box trucks that move it to the local PO are also owned by private companies.
The PO has been outsourcing their work to non-union shops for quite some time.
Several of the staples stores in my area aleready ship packages via UPS.
“I wonder if UPS, FedEx or someone else will take their place?”
They are doing it already.
FedEx Smart Post and UPS mail. They handle post mail packages part way and deliver to the local post office for home delivery.