1964 Plymouth Valiant-Coolant Issue

I recently purchased a 1964 Plymouth Valiant with a 318. After having a bit of work done, it was evident that the car was sitting for some time (brake fluid congealed, gas smelled/looked foul, etc). Temp gauge was faulty, so I replaced the sensor. After doing so, noticed temperature was on the high side of the factory gauge, so I had a 160 degree thermostat put in. Still runs hot. Got it to running temp the other day in driveway and took the radiator cap off. No flow/circulation of coolant at all. Coolant is green, just doesn't move anywhere. For now, I'm assuming the thermostat works. I am also assuming that given the evidence of long-term storage, the radiator could benefit from a good flushing. My quandary: If I decide to flush, what is the danger of possible motor damage (ie freeze plugs/head gasket leaks, etc)? Would I be better served installing a new radiator and avoid what could be the beginning of many problems, or is there something more obvious I should be addressing?