tl;dr  For the first millisecond oil is faster, after that water is faster.

The specific heat capacity of water is about 1 cal per gram per degree C.

The specific heat capacity of oil is about half that.

The latent heat of vaporization of water is about 540 calories per gram.

The latent heat of vaporization for oil is about 60 calories per gram.

The latent heat of vaporization of water is about 540 calories per gram.

The latent heat of vaporization of oil  is about 60 calories per gram.

The average boiling point of water is around 100°C

The average boiling point of oil is around 200°C

The density of water is 1, and that of oil is less, call it 0.9.

Each cc of water can absorb 100 cal to get from 0 C to 100 C, and then 540 to become water vapor.   That's 640 cal per cc.

Each cc of oil can absorb 90 cal to get from 20 C to 200 C and then 60 to get to vapor, but multiply by the density.   That's 0.9*150 = 135 cal per cc.

So a bucket of ice water can absorb about 4.7 times the heat that a bucket of oil can.

But what about speed?

It is true that for a short time, the hot steel in water will be surrounded by a thin layer of steam making the boundary temperature just outside the steel 100 C or even more as heat transfers from the metal to the steam.  And this layer insulates the metal from the liquid water. But simultaneously, the outer most layer of metal is becoming cooler than the center of the solid metal so inside the metal, a temperature gradient develops.  Also the >100 degree C steam boundary layer is rapidly removed and replaced by 100 degree steam.  All this steam turnover is removing 640 cal for each gram of 0 degree water converted.

So in water, the temperature just outside the metal is initially 0 C; for maybe a microsecond.  After that it is 100 C as long as there is boiling.

In oil, that temperature starts at 20 C; for a microsecond.  Then it rises rapidly to 200 C in maybe milliseconds.

So for an extremely short time, the oil cools the metal faster (maybe for a millisecond) and after that the water cools it much faster.

If we are assuming there is no agitation and the boundary layer of vaporized water is stable, then we must also assume that the boundary layer of oil is not immediately swept away either.