Absentee ballots can be sent with private couriers like FedEx. That's actually what some people living overseas have to do. Election commissions can choose to send ballots via private mail couriers as well, and allow people to print and send self-certified ballots.
How much does that cos? The post office is cheap for a lightweight letter (and I can use a stamp from home), but if I use FedEx I generally have to go to their local shop and pay a few dollars
"And we need a post office to do that."
No you don't, you just need some kind of mail delivery service with either reasonable rates, or 'paid for' by dear government.