The army Landrovers I drove, back in the day, all had normal (thin, high profile) tyres on large wheels. For that very reason. As long as the mud was not deeper than the ground clearance she would usually plough through.
I can't recall, did the Dodge film have any climbing in the footage?
The other things to consider: If they smashed the springs, or otherwise damaged the car, the footage would have been removed from the promotional film. These old cars were solid, which is why in a crash (even at low speed) the passengers always came off worst.