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.

There are various figures of merit, such as the Mouromtseff number https://doi.org/10.1109/JRPROC.1942.234654 or https://www.electronics-cooling.com/2006/05/comparing-heat-t... for a quick overview.

Some tables of heat transfer coefficients: https://www.engineersedge.com/heat_transfer/convective_heat_...

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.