Early biplanes did use similar airfoils. Not as extreme as the Eppler 376, but still very thin and highly cambered.
When Otto Lilienthal started his glider experiments, he tried to copy storks. He experimented with different airfoil shapes by using exchangeable ribs on the gliders and by testing model wings on a rotation test stand (Rundlaufapparat). There he discovered that thick airfoils with a blunt nose were actually better than the thin, birdlike airfoils he had used so far. But he didn't believe his own results and continued with birdlike airfoils.
Farman-Voisin biplane, 1907 (picture source).
The same happened with all other airplane designers until 1915, when scientific work started to influence airplane designs. The highly cambered airfoil works very well in a very small range of angles of attack when the local direction of flow is parallel to the local contour of its nose. But while birds can adjust the camber and area of their wings, airplanes of those times could not. In order to combine fast flight with high lift for take-off and landing, the thick airfoil is better.
While birds are small enough to structurally get away with thin wings, the much larger man-carrying airplanes need the thicker wings to accommodate their bending loads without bracing. Scaling laws show that loads grow faster with size than the dimensions do and only thicker wings make the unbraced, cantilever designs necessary for efficient transportation possible.
Heron in flight (picture source). This picture shows nicely that 90% of the area is feathers, so a thick wing becomes impossible for birds.
Birds cannot be hollow inside, except for hollow bones. So they have no choice of using thick wings - they have to work with thin wings and adjust camber and wing area to the flight conditions. Being much larger, airplanes need thick wings for aerodynamic and structural efficiency.