Babinsky, Holger (2003) are absolutely correct in pointing out that overpressure underneath the wing will contribute to lift.
However, this is not necessarily related to "thickness", which is more related to strength.
Scale is extremely important in aviation design: to support its own weight, scaling a sparrow up to a Piper Cub requires a wing that can support several thousand kg (don't forget manuvering G forces). This will no doubt require a proportionally thicker wing$^1$.
As mentioned in the above reference, curvature, or camber, determines lift at a given speed, wing area, Angle of Attack, aspect ratio, and air density.
At first glance, the thin wing would have the advantage of creating over pressure under the wing to enhance lift, but thicker wings in even the largest planes also have this trick up their sleeves by using ... slats and flaps for lower speed flight. The "thick" wing will also have curvature on its upper surface, which leads to another very important consideration:
Reynolds Number = Velocity × Chord/Kinematic Viscosity
Kinematic viscosity for air is 1.46 × 10$^-5$, the units are meters and seconds
This will lead to a better understanding why larger, faster aircraft fly more efficiently using "top lift". A look at Reynolds number vs Lift/Drag ratios on Airfoiltools will show a marked increase in L/D ratios as Reynolds numbers rise from 10$^3$-10$^4$ typical of birds to 10$^6$-10$^7$ typical of aircraft.
Larger, faster aircraft can streamline the bottom of the wing, adding strength and reducing drag, by relying on more efficient "top lift".
Here, the legend of the "thick wing" was born, exemplified by fighter designs of the Luftstreitkrafte and the Davis wing years hence. Thickness and curvature also become relevant as velocity approaches the trans-sonic and super-sonic realms (critical Mach number), where shockwave drag effects favor thinner wings with less camber.
Bird wings are generally too small and slow to take full advantage of the famed "Bernoulli" top lift effect. One may realize a diving hawk may cross into higher Reynolds numbers, but they solve this issue with ... variable geometry, by folding their wings!
$^1$ early aircraft builders supported their beloved thin
undercambered wings with cables, and by stacking the wings
(biplane), created very strong, light truss structures.
Emphasis on reducing drag would come later with higher
speeds, enabled by more powerful motors.