As ezrast says below, the real goal is a "reliable communications network", which I agree with wholeheartedly. In the 18th century, that was spelled "Post Offices and Post Roads". They couldn't have anticipated email. Must we interpret the Constitution's requirement so literally?
Maybe "post offices and post roads" could mean "internet service". We're seeing public school taught over the internet today, due to COVID-19, so I think it's fair to call the internet an essential service. That's how I receive 99% of my communications. Does even rural Montana really find it more valuable to send pieces of paper than internet packets?
Let's stop printing 100 million advertisements every day just to drive them across the country and throw them away somewhere else. Instead, let's use the office to provide internet access to everyone. I think that would be exactly in line with the spirit of the Constitution.
USPS lost $9 billion in 2019[1], so roughly $30 per American. Anyone who pays even one bill per month via mail would pay at least 4x that much at current private rates. I'm not inclined to believe privatization would bring those rates down much for rural areas.
Sure, if we assume we've implemented UBI and eradicated poverty or something then those prices aren't a big deal. But even then, what have we gained by privatizing? Private industries that provides critical services are either massively subsidized (agriculture, transportation - indirectly via roads) or massively unpopular and frequently monopolistic (telecommunications, health insurance). Even if a talented CEO could squeeze a few percentage points of efficiency out of this industry, it would come at the cost of misaligning incentives in perpetuity, and the government would still be responsible for keeping the industry afloat whenever the economy goes bad.
Or we could let them charge a few extra cents for postage, let them operate 5 days/week instead of 6, charge the wealthiest Americans a couple extra tax dollars each, and keep consistent service available at dirt-cheap prices for everyone everywhere.
[1] https://about.usps.com/what/financials/annual-reports/fy2019...