Is it specific heat that we care about, or rate of heat transfer?
Specific heat matters a bit, but if you make your coolant take twice the energy to change 1 degree, the same thing happens on the radiator side and you must release twice the heat to cool 1 degree.
Rate of heat transfer in general if probably more important.
Well, it gets really complex. Yes, specific heat matters, but as you say so does the heat transfer coefficient. And the viscosity. And is the flow laminar or turbulent? Etc.
But, turns out water is just very very good also when you take these other factors into account. Compared to oil, it has, as mentioned, much higher specific heat, it has higher heat conductivity, it has lower viscosity which means less pumping power and more likely to see turbulent flow which helps with mixing.
In the case of the oil pull tractor they needed to run the tractor much hotter than water boils - even if they had known modern antifreeze it still boils too hot. They also didn't know to make modern radiators - they usee very large radiators to make up for lack of surface area. (They also used exhaust to pull air through the radiator instead of a fan). All in all a very ingenious design - but there is good reason they don't make them like that anymore. (My 1939 is a lot more modern - and it lacks electric start)
Rate of heat transfer in general if probably more important.