Persons is a set of individuals while people generally implies a group. "Those people" as a phrase carries negative connotations because of its use by racists pretending to be not quite as racist.
You can use them interchangeably as long as you're trying to imply the person's you are referring to are linked or similar in some way.
Your math isn't mathing. He isn't talking rank and file Republicans. He's just talking about their leadership and policymakers. That's totally different. Mr Trump can be friends with Epstein all day long and that couldn't possibly reflect poorly on the rest of the republican party. He's only their elected nominee for president.
Get your "not all men" horseshit out of here.