It's basic science - many flammable liquids are at at temperature of about 50C when ignited, whereas cooking oils are ignited by being heated to their fire point where they auto ignite - this is around 350/360C
This temperature defeats most agents:
Foam - Even on normal liquids foam blankets are destroyed by heat, but the extreme heat of cooking oils destroys the foam so fast that it can't form a stable blanket, especially with modern film forming foams applied non aspirated (resulting in a thinner blanket)
CO2 - No cooling or physical blanketing, disperses so cannot reduce the oxygen supply long enough
ABC Powder - The inhibitory effect of the powder will knockdown flames, but when the extinguisher is empty there is still enough heat in the oil for it to restrike and no physical blanket to cut off the oxygen
Effective agents work in varying ways:
Wet Chemical - The water based spray/mist discharge cools the oil whilst the potassium salts in the solution cause saponification of the oil into a soapy sludge that dries out and crusts due to the heat creating a physical barrier between the remaining oil and the oxygen around. The potassium salts also have inhibitory effect hence the almost instant knockdown of the flames
Class F foams - Phosphoric acid salts are added to the foam concentrate to stabilise the foam blanket against the heat so it can smother the fire also allowing the water content to vapourise and cool the oil
Water Mist - The fine mist turns to stream before mixing with the oil so there is no ejection of oil, the fine water particles with their large surface area absorb heat quickly, allowing the oil to cool below ignition temperature.
Some videos -
https://www.youtube.com/watch?v=0IqM2dvl7Dg&ab_channel=FireWizard &
https://www.youtube.com/watch?v=fYlQDAyR-ME&t=1s&ab_channel=FireWizard
