I wonder, did they provide a reason for the lack of a direct relation?
The gist seems to be that airspeeds vary a lot less than wing loading varies—the range of airspeeds is 'compressed'—because other things affect the speed of birds' flight, too. One such thing is their wings' aspect ratio: as wing load increases, wings tend to become deeper, from leading to trailing edge, and wingspan tends to decrease.
Obviously another factor is the cadence at which birds flap. Loons and sea ducks with high wing loading (and high aspect ratios) flap quickly and don't attempt to soar. Partly, as an aerodynamic matter, they have to if they hope to stay airborne; and partly they may just want to move faster.
Different families have different wing designs, and the specific details of those designs—in other words, what they're evolutionarily best adapted for, whether it's soaring, navigating through brush, or quartering, for instance—has a lot to say about the speeds they usually achieve.
As the authors sum it up, in language that is only slightly too elevated, "functional flight adaptations and constraints associated with different evolutionary lineages have an important influence on cruising flapping flight speed that goes beyond the general aerodynamic scaling effects of mass and wing loading." This is commonsensical enough that I wonder a little whether they framed their article to refute a straw man.
As an unscientific observer, I am interested not so much in airspeeds as in the way birds look when flying, and what sorts of flight they are most comfortable with. Wing loading is only one number among many, but it does explain some of this variation, and I think helps me to understand flight styles.
For instance: the online edition of Birds of North America notes that male Northern Harriers "have shorter wings and lower wing-loading than females … and appear more agile in flight." Partly because of lower wing loading, males burn only about 4.7 Watts in flight compared to females' 5.6 Watts, and have been observed outclimbing Prairie Falcons.
Male Short-eared Owls, which have a wing-loading factor of 0.281 g/cm^2 (females' wing loading is 0.385 g/cm^2; David H. Johnson, "Wing Loading in 15 Species of North American Owls," paper presented at Biology and Conservation of Owls of the Northern Hemisphere symposium, February 5–9, 1997), have famously 'buoyant' flight, and are able to outclimb pretty much any bird they meet.
So there's more to life to airspeed, even if wing loading can't be treated as a universal explanation for the ways birds fly.
P.S. Given what you say, I'm all but guaranteed to go down a Wright Brothers rabbit hole next! Thanks for mentioning their book.