The author seems to over-state the difficulty of walking, which is surely no greater biomechanical challenge than the flight of a raptor or the dexterity of a gibbon. We humans always tend to self-aggrandize, blowing our own accomplishments up because we fail to calibrate adequately against other species. The size of our brain is barely related to our mode of locomotion (raptors, whose brains must perform many more calculations in a much shorter period of time, are hardly enormous).
There's an interesting hypothesis that notes our ancestors gradually adopted an increasingly bipedal lifestyle just as grasses were eroding the dominance of trees. Grass uses fire as its weapon (grass burns hot, killing trees, but burns quickly so its roots remain largely intact, permitting rapid regeneration while the trees remain burned-out hulks). So there was an evolutionary niche to be exploited and our ancestors happened to be the primate species that most easily adapted itself to this new tree-reduced environment.
As we did so, we very slowly increased our tool use (other primates use tools, but ours are the most sophisticated) because our hands were now freed up for activities other than simply getting from Point A to Point B. And so the benefits of bipedalism were compounded, giving our precarious species slightly better odds of survival.
As for the argument that "you don't want to stick your head up to see predators" we need only note that armchair anthropology is rarely a reliable guide. Predators were always faster than our ancestors and any advanced warning would have been better than none. Think of all the prey species that rely on alarm calls to alert the group, even as the maker of the alarm call incurs slightly greater personal risk at times.
In short, while we do not know the details, we can certainly discount most of the naive hypotheses and focus instead on those that are more consistent with what evidence we have. As such, we may regard ourselves as children of grass not just because of our reliance on grass (wheat, rice, barley, etc.) since the agricultural revolution but because grasses created the very ecological trigger that brought our distant ancestors out of a largely tree-centric lifestyle.