This logic could be applied to literally anything, so your argument is effectively that the government should never fund anything.
If there is a war, cancer/disease research is going to be less important, so the government shouldn't fund cancer/disease research.
If suddenly a famine strikes, war is going to be less important, so the government shouldn't fund the military.
If a sudden deadly disease arises, funding for food security/research is going to be less important, so the government shouldn't be funding any of that as well.
But a mail and parcel service, something that the private sector does profitably, shouldn't be deeply in the red though a little from time to time is probably fine.
What's causing the unlevel playing field?
Although I thought USPS had an enforced monopoly on US mail, so how did you do the comparison with private sector mail?
[0] https://about.usps.com/newsroom/national-releases/2019/1114-...
The USPS has a monopoly on traditional delivery to a mailbox. So FedEx walks all the way to the door which makes it not "traditional delivery"
Sometimes I find Amazon packages in my mailbox, though I know sometimes amazon will use USPS, so could just be them.
Thankfully, the government guarantees it will deliver letters to some remote rural places at a price private companies can't touch, but we can do better to make life easier for everyone: the mail man, the people wanting their mail, etc.