Sometimes the veins in the radiator get plugged. Flushing and back flushing may help a little but is better to get them cleaned out by a professional. I had to go around to a dozen radiator shops before I could find a shop to remove the tanks and clean the veins out. Everyone wanted to re-core it and I would loose the hole for the crank. Like you I wanted to keep the original look so with doing the clean out and swapping out the original fan with a GT6+ metal fan it has never over heated. The GT fan looks original and after I painted the blades silver no one has ever noticed it even in a LBC show.
They don't even have to be plugged all the way. Last time I went around this tree, I was using a radiator that had worked fine in the past but sat unused for a couple of years. The radiator shop said it flowed fine, there was no need to rod it out. After trying everything else, I finally insisted that they do rod it out. Surprise, surprise! Although none of the tubes were actually blocked, they had a terrible time forcing the rods through, and wound up with 5 or 6 leaks once they did. The inside of the tubes were coated with "mud" (highly technical term there, doctor) that was apparently mostly old stop-leak; and was blocking the transfer of heat from the coolant to the fins.
Rather than mess with the leaky core, I had a new core installed. My shop gave me the option of putting a hole in the new core, but said it would reduce cooling capacity by about 10%, so I elected no hole. All of my cooling problems vanished like magic!