Actually, it is the other way around. Dry air has less heat capacity than humid air so it removes less heat than humid air at the same temperature.
Air has a specific heat capacity of slightly more than 1kJ/kgk at room temperature, but humid air (say 60% RH) has a specific heat capacity of 1.8 kJ/kgk - so it can remove almost twice as much energy (per unit mass) than the dry air when flowing across a radiator. Think about it as something more dense that can suck up more heat.
It’s the human body that relies on sweat evaporation to cool itself that is adversely affected by humid air, not hot surfaces exchanging heat.