The patent was pursued by National Foundation for Infantile Paralysis who funded Salk's research.

Salk himself wasn't much in favour of a patent, stating that his techniques weren’t novel and the work was built upon the research of others.

https://www.latimes.com/business/hiltzik/la-fi-mh-polio-vacc...

All of those famous public statements by Salk occurred after his work was determined to be unpatentable by the funding agency. We don't really know what happened behind closed doors other than the normal university/nonprofit research process, which involves a lot of profiteering. The same thing happened with Sabin before that vaccine was also released without a patent (Sabin would have been a Soviet/Russian patent, so it's a slightly different system).

The famous "could you patent the sun" quote was about obviousness, not about anything else.

This runs counter to every source I've read - do you have a source for it?