The example you gave doesn’t make sense. First off you confused public trading (company shares are available to the general public) with public ownership (owned by the government i.e. “the public” at large). Johnson and Johnson is publicly traded but the shares are held by private entities. If I buy a share of Johnson and Johnson’s stock, I privately own a piece of Johnson and Johnson.
As for drug patents (and patents in general), the idea is to secure timed exclusivity to sell in the market in exchange for public disclosure of method of invention. If we didn’t have patents, companies would instead treat drug formulations as trade secrets and so they’d hold onto that exclusivity as long as they can keep the formulation a secret or until another entity reinvents the same thing. There are issues with the patent process and especially with private companies benefiting from publicly-funded research while locking up exclusivity and jacking up prices, but those are still problems with capitalism, and they’re still better than just letting the free market completely monopolize the process.