In the Sonoran desert (which is very much not a fire-adapted ecosystem) the cheno-ams provide a flush of greens right after the monsoon rains arrive. No burning required...in fact, prior to the introduction of Old World grasses to the region, there really wasn't a way to burn the desert. The higher elevation grasslands burned every summer at the outset of the monsoon thanks to dry thunderstorms.
Native peoples in the lower deserts harvested cactus
fruit and mesquite beans in the hottest part of the summer, and then when the rains arrived they got the burst of amaranth and lambs quarters to keep everyone fed until their floodwater crops were coming along.
The Spanish padres who arrived in the Pimeria Alta in the 17th century were horrified to see people happily eating the greens on all fours, grazing like sheep, and thought it was evidence of how far they had fallen from grace. The reality is more like why would you waste time and effort hand-picking your salad when it's all laid out in front of you? I used to love seeing the cheno-am carpet on the fallow beds in my garden in the middle of summer, and tried the traditional eating style myself.