Formerly on the NALC website (and saved by the Wayback Machine:
Although the NALC objected strenuously to one provision—requiring injured postal employees to wait three days before beginning Continuation of Pay benefits—the union played a crucial role in developing many of its most important provisions.
(Note that that one provision is something no one remembers or cares about today.)
The Constitution had to explicitly allow the government to deliver mail since otherwise the monopoly would have been illegal. However, there is no requirement for it either (unlike the original Articles of Confederation which explicitly gave the government a monopoly on mail delivery) so the government is free to get out of the mail delivery business if it chooses to, without amending the Constitution. Hence the fact that no privatization attempt has ever included a proposal to amend the Constitution, since it's unnecessary. If there actually was such a requirement, postal unions wouldn't waste time opposing privatization attempts - they'd simply ignore them, knowing they'd be struck down.