If you watch full, you will notice how tall and thick plants or dead wood and metal parts this guy was passing through, esp here.
I strongly suspect that some bushes, metal or wood piece burst the radiator bottom hose just before he stopped and switch off the engine. Initial smoke seems like when you drive without coolant and engine is tossed, seen this exact thick white smoke before a couple of times just before car catches the fire.
Though this guy looks very calm, compose and organized throughout the disaster and said he used two fire extinguisher and 2 gallons of water. What I suspect is he wasted the fire extinguisher on flames rather than the base of the water and then 2 gallons water help spread the fire in the engine bay to the other heated parts.
In such a scenario, even if he would have closed the hood of the engine there are very strong chances he would have saved from this disaster as by cutting the oxygen alone you can put off the fire.
Anyways quite a few lessons for all us here:
Don't cross thick tall plants and bushes, unless your car underbody is fully protected with bash and skid plates
Never drive alone, at least go with 2-3 cars in any off-road terrain.
Keep fire extinguishers handy and always use with a calm and focused approach to point them at the base of the fire.
For engine bay fire, putting sand inside the engine bay is better than water to not to spread the fire from leaking fluid that has already caught fire - fuel, engine oil, brake oil, gear oil etc.
Keep your emergency items handy to escape in case you need to leave the vehicle suddenly.
Always remember to grab your wallet and phone to call for help and deal through the situation.