It's hard for me to second guess but I think if it had been me, as soon as the car got stuck I would be backtracking the way I came in. If a car made it in, it's clearly going to be pretty easily walkable, and you're heading back towards more main roads where a passing car is more likely.
I can't imagine taking off into the brush with no water or gear. Especially with kids.
The author did an excellent job of getting into their heads in the situation and rationalizing their decisions. They underestimated road conditions; "their pamphlet offered an (erroneous) alternative, apparently shorter route back down to the valley in the form of a road down Anvil Canyon;" they made an easy-to-make wrong turn; and they were operating under an assumption that help was a few miles south instead of dozens of miles back.
"Given the information available to them, where could they best find it? It is likely they had gone the entire day without seeing another human, so their entry route offered little hope of others. In examining their maps, Ballarat was many miles away, and with unknown services. Furnace Creek, with everything they needed, was far to the north, out of their reach. Even the paved Badwater Road on the valley’s east side had only intermittent traffic and was in a zone of lethal heat. But there was another possibility... [only about 8 or 9 miles to the south of them]"
I can't imagine taking off into the brush with no water or gear. Especially with kids.