Beaks are awesome tools. I have trouble believing that eating competitions result in either large mammal preeminence, or the greater diversity in small birds. From what I've read I think you would agree with this. Like you I would be interested in hearing how any competitor in this respect could affect ostrich distribution. Indeed, ostrich adults living among the world's most formidable carnivores suffer little predation.
Winged flight enables a bird to forage and then escape to cover.
For example, a small mammal out of cover risks more than a similarly-sized bird. The bird can fly and maneuver away from, say, aerial attack. Small animals have more hiding places. But these hiding places may not occur where there is forage. The greater range of birds allows them to both exploit concealment and remote patches.