He makes the same point you missed when you misquoted his discussion about the evolution of the eye (the incomplete parts of the quote you provided are bolded):
In any case, I introduced (gliding) flying fish as a prelude to the theory that true, flapping flight evolved not from tree gliders but from fast-running, ground-dwelling animals whose arms became freed from their normal role in running. Flying fish and flying squids, although they live in water, illustrate the principle that if a gliding animal can move sufficiently fast along the surface it can take off without the support of a tree or cliff. The principle might work for birds, because they evolved from two-legged dinosaurs (indeed, you could say that technically birds are dinosaurs), some of whom probably ran very fast along the ground, as ostriches do today. To pursue the analogy with flying fish for a moment, the two legs would play the role of the fishs tail, propelling the animal forward very fast, while the arms play the role of fins, perhaps originally used for stabilizing or steering, and later growing aerofoil surfaces. There are some mammals such as kangaroos that propel themselves very fast on two legs, leaving their arms free to evolve in other directions. Our species seems to be the only mammal to use the two legs in the alternating, bird-like gait, but we are not very fast and we use our arms, not for flying but for carrying things and making things. All the fast-running, twolegged mammals use the kangaroo gait in which the two legs push together rather than alternately. This gait grows naturally out of the horizontal spine-flexing of a typical running quadruped such as a dog. (By analogy, whales and dolphins swim by bending the spine up-and-down, mammal style, whereas fish and crocodiles swim by bending it alternately to left and to right, following the ancient fish habit. Incidentally, we should wonder more than we do at the unsung heroes among the mammal-like reptiles who pioneered the up-and-down gait that we now admire in sprinting cheetahs and greyhounds. Vestiges of the ancient fish wriggle are perhaps still to be seen in tail-wagging dogs, especially when the movement spills over to the whole body in the squirming of a submissive dog.)
Among ground-dwelling mammals, kangaroos and their marsupial kind dont have a monopoly of the kangaroo gait. My colleague Dr. Stephen Cobb was once lecturing to zoology students in the University of Nairobi and he told them that all wallabies are confined to Australia and New Guinea. No Sir, a student protested. I have seen one in Kenya. What the student had seen was undoubtedly one of these (Figure 4.5).
This animal, the so-called springhaas or spring hare, is neither a hare nor a kangaroo but a rodent. Like kangaroos, it hops to increase its speed when fleeing from predators. Other rodents like the jumping jerboa do the same. But piped mammals dont seem to have taken the next step and evolved the power of flight. The only true flying mammals are bats, and their wing membrane incorporates the back legs as well as the arms. It is hard to see how such a leg-encumbering wing could have evolved by the fast-running route. The same is true of pterosaurs. My guess is that both bats and pterosaurs evolved flight by gliding downwards from trees or cliffs. Their ancestors, at one stage, might have looked a little like colugos.
Birds could be another matter. Their story is different anyway, centred around that wondrous device, the feather. Feathers are modified reptilian scales. It is possible that they originally evolved for a different purpose for which they are still very important, heat insulation. At all events, they are made of a horny material which is capable of forming light, flat, flexible yet stiff flight surfaces. Bird wings are very unlike the saggy skin flaps of bats and pterosaurs. The ancestors of birds therefore were capable of forming a proper wing which didnt have to be stretched between bones. It was enough to have a bony arm at the front. The stiffness of the feathers themselves took care of the rest. The back legs could be left free to run. Far from being awkward and clumsy on the ground like bats and, presumably, pterosaurs, birds can use their legs for running, jumping, perching, climbing, prey-catching and fighting. Parrots even use their feet like human hands. Meanwhile the front limbs get on with the business of flying.
Heres one guess at how flying got started in birds. The hypothetical ancestor, which we can imagine as a small, agile dinosaur, runs fast after insects, leaping in the air with its powerful hind legs and snapping at the prey. Insects had evolved into the air long before. A flying insect is perfectly capable of taking evasive action, and the leaping predator would benefit from skill in mid-course correction. To some extent you can see cats doing this today. It seems difficult because, since you are in the air, there is nothing solid to push against. The trick is to shift your centre of gravity. You can do this by moving bits or yourself relative to other bits. You could move your head or tail, but the obvious bits to move are the arms. Now, once the arms are being moved for this purpose, they become more effective at it if they develop surfaces to catch the air. It has also been suggested that the feathers on the arms originally developed as a kind of net for catching insects. This is not so far-fetched as it sounds, for some bats use their wings in this way. But, according to this theory, the most important use of the arms was for steerage and control. Some calculations suggest that the most appropriate arm movements for controlling pitching and rolling in a leap would actually resemble rudimentary flapping movements.
The running, jumping and mid-course correction theory when compared with the tree-gliding theory, reverses the order of things. On the tree-gliding theory, the original role of the proto-wings was to provide lift. Only later were they used for control, and then finally flapping. On the jumping for insects theory, control came first, and only later were the arms with their surfaces commandeered to provide lift. The beauty of this theory is that the same nervous circuits as were used to control the centre of gravity in the jumping ancestor would, rather effortlessly, have lent themselves to controlling the flight surfaces later in the evolutionary story. Perhaps birds began flying by leaping off the ground, while bats began by gliding out of trees. Or perhaps birds too began by gliding out of trees. The debate continues.
-Richard Dawkins, Climbing Mount Improbablepp. 122-126