I sort of suspected you might find a dirty radiator core. Once a layer of hardened crud accumulates inside the tubes, it forms a very effective insulator that prevents the proper heat exchange from taking place. Even the acid flush I do can fail at times, if the stuff is thick enough, or the chemical composition of the deposits resists reacting to the acid. Good luck on the pump rebuild!
Even though your radiator recore job is rated for higher pressure, be aware that the heater core probably isn't. Be cautious about using a cap that's compatible with the radiator, not the heater.
Jerry