An INTERPOL arrest warrant can only be executed by the authority in the jurisdiction where the "fugitive" is located. For example, if the U.S. issues an arrest warrant for a suspect in the U.K., Scotland Yard executes it regardless of whether an INTERPOL arrest warrant exists or not. In fact, seeing that the U.S. and the U.K. are signatories to another threaty (whose name escapes me at the moment), there would be no reason add another layer of paperwork, and the U.S. and the U.K. will likely work things out on their own.
INTERPOL arrest warrants are used when the suspect is "country-hopping" to avoid the law, or when one of the states involved is not a signatory to an extradition treaty with the other. In other words, say the suspect is believed to be in the U.K., but may have fled to France. Instead of waiting for the suspect to be located before an arrest warrant can be issued in the local jurisdiction, an INTERPOL warrant is used as a spring-board to get the process rolling, by providing a centralized location/database where the two (or more) law enforcement agencies can cooperate.
In your italicized hypothetical above (I'd love to hear you identify a case where it has actually happened), presumably you speak of a case where an American suspect is under an arrest warrant from [insert banana-republic hellhole here]. In such a case, even if that hellhole managed to get an INTERPOL warrant issued (not easy, since INTERPOL assiduously protects its independence for the reason it does not wish to be seen as an arm of the "hellhole police"), INTERPOL itself would be powerless until the U.S. executed the warrant.
In sum, your hypothetical U.S. citizen, if he is arrested and extradited, has a beef with the U.S. Departments of Justice and State, and can take his dispute to U.S. Federal Court (and ultimately the U.S. Supreme Court), most likely the one that issued the U.S. arrest warrant to initiate the process. Thank you for letting me share.
I must admit, the image of uniformed INTERPOL officers executing warrants in the U.S. independently of U.S. law enforcement is rather humorous.