I claim your thesis has a critical corollary that goes beyond "competitors would take away part of USPS's marketshare if they were allowed to compete". Specifically that the competitors would cherry pick the most profitable business of the Post Office and force it out of business leaving much of the nation unserved, or into a business model we don't think would be good for our society (first class mail to Eagle Butte, SD costing, say, $10/oz).

There's also friction issues. If I'm going to send out every month a set of bills to my customers, I know the cost will be X customers * the rate I pay (which I can decrease by making things cheaper for the USPS),