Flat tires are the only thing I recall being a real annoyance. But, I still get one once in a while and AAA takes care of it. I don't remember any of our 50's & 60's cars overheating. My Model A would overheat sometimes when climbing
We had a '77 Plymouth Volare, with the slant 6, that we took on a trip through Death Valley in the summer of '80. It started to run hot to the point of almost overheating but we had the A/C running. It was easily over a 100 outside at the time. We had to turn the A/C off and drive with the windows closed since it felt like a blast furnace with them open.