How long does radiant heat in a concrete floor last?
According to the Copper Development Association, it is perfectly fine to embed copper tube in concrete under certain conditions. However, it is unlikely your home met all the conditions like making provision for thermal expansion (especially important with a heating system that will likely cycle on and off many times in one day) Another important requirement is to protect the copper if "it comes in contact with concrete mixtures that contain components high in sulfur, such as cinders and fly-ash, which can create an acid that is highly corrosive to most metals including copper."

In the early 80s I lived in a house that had plumbing in the concrete slab floor. Not heat, just domestic water. It was much younger at the time than your house is now. There was a leak and the only way to find and fix it was to jack-hammer the living room floor. Because the water followed existing cracks in the concrete, a lot of concrete had to be removed to find the actual location of the leak.

My recommendation: write off the radiant heat system and budget for a forced air heat replacement. You can upgrade the house to have air conditioning for the summer at the same time.