Trump correctly points out that consumers already choose between private package deliverers. However, the USPS also faces crushing competition from digital forms of correspondence. USPS mail volume has declined steadily, and postal employment has fallen from 900k in 1999 to 600k now. This decline coincides with steady population growth. The truth here is that the USPS continues to function for two main reasons: the nostalgic value of traditional letter deliveries to some, and stalwart support for every bit of the public sector by ideologues.
Looks like there are a lot of reasons to go private and a few to stay as a public service. The solution may be to just let anybody deliver the post (letters) that wants to do that business. Then let the USPS live or die by competition.