From Wikipedia[1]: Article I, Section 8, Clause 7 of the United States Constitution, known as the Postal Clause or the Postal Power, empowers Congress "To establish Post Offices and post Roads".
Without this clause, Congress would not have the authority to establish Post Offices. But note the clause does not say anything about making postal service an exclusive government function. In fact the clause does not even mandate that Congress establish Post Offices, Congress merely has the power to do so. If Congress shuts down the Post Office, it would be perfectly Constitutional. The clause is definitely not a "reason for subsidizing rural mail delivery" anymore than Congress's power to Declare War is a reason to have a war.
The Founding Fathers considered postal delivery to be important enough to mandate it in the Constitution.