Does any company want to ship letters for that matter?
Impossible to know unless you give people the option. Lysander Spooner certainly wanted to ship first class mail, as did William Dockwra and Robert Murray in London.
"Impossible to know"
No its not. The ceo of UPS and fedex could say at anytime they are very interested in being able to ship letters. Have they ever said that?
Have they ever expressed interest in changing the law that forbids them?
Your question is so specific as to hang on the wills of the CEOs of these two companies. Depending on their knowledge of the market place and their cost structures, both of which are subjective, the individuals have chosen at this point not to beg permission from government to deliver first class mail.
This is not the same as allowing any individual out of the other 5.999999 billion people to deliver the mail. Who knows what competitors to the USPS would come up with given the opportunity to do so.
The same is true of any cartelized business. The sole purpose of these cartels is to restrict competition.