I think you are on the right track. It wants/needs a shroud for low speed cooling performance. That is first step.

The radiator could be getting plugged as well. In my experience, parts store radiators for classics can be weak to start with and some deterioration will make them non functional. Another check would be the fan clutch. Inspect for loose/sloppy and feel for air flow in front of the radiator are the best checks I know.