Cooling systems can become a monster in waiting if the internals are gummed up with sludge and contaminates that can occur on older machines; especially if any of the previous owners used the wrong (cheap?) coolant.
To be sure of a fully functioning cooling system, renew all the various parts that may now be terminally blocked or suffering internal corrosion.
This item and some of the parts below may be required to achieve a fully successful overhaul of the radiator and it's associated parts.
The more parts in the cooling system that are renewed the less potential problems incurred. Add as many items as you wish the same shopping cart, new fasteners, hoses and hose clips for example, and still only pay the same fixed price shipping cost.