Removing the patent doesn't save more people, just as many vaccines would get produced, it just shifts money from the company to the state that pays for the vaccine.
Money and risk is a very material part of the equation, now that the dynamics have changed, a lot more speculative money will be available. Supply and Demand works on cash just as it does for other things.
Also I don't think it has a populist aspect.
Exporting 'already made vaccines' does have a populist risk - a lot of Americans may 'get really mad!' if the US is seen to be exporting vaccines to other places before the US is fully vaccinated, irrespective of whether their chagrin is misplaced or not, the news cycle populism may have a field day with that. And there's the contractual stuff which he cannot 'waive'.
Second, it's a lot to do with the 'rest of the world' not really the US.
Third, AZ was UK not Europe, J&J was Neterhlands but owned by an American firm, Pfizer/BioNTech was both, Moderna was USA. So I don't see how this is a 'Europe victim' scenario.
Exporting individual vaccines might break contracts, and because it's a little bit zero-sum, the public at large may hold him in contempt for not putting 'America first'.