If you ran Dexcool you'd find out right away thanks to the rust colored shit it's made with. My Camaro was overheating because the aftermarket efans weren't coming on. The boiled coolant would come out of the overflow.

Also, when i replaced my radiator exploded and I had to replace it with an aluminum Summit 4 core, it looked like a shotgun blast of diarrhea in my engine bay. Fortunately the gel coat was still on under the hood so that wiped right off.

I think you're right, though. Top it off, get it hot, and look for leaks.