Why doesn't an airfoil rotate?
It does rotate. In fact, in aerodynamics, this rotation force has a name: pitching moment. Pitching moment changes depending on the angle of attack.

Airfoils can be designed with almost any pitching moment and still generate lift, from positive (trailing edge rotates upwards as you describe), to zero (no rotation) to negative (trailing edge rotates downwards). In general, high lift airfoils have positive pitching moment (because, as you noticed, downwash caused by the Coanda effect increases lift). But zero and negative pitching moment airfoils are useful for tailless and flying wing aircraft. Helicopter blades are long, narrow and flexible so must use zero pitching moment airfoils to prevent them twisting.

If you use an airfoil with a positive pitching moment you need something to counter the rotation. For most aircraft the solution is the horizontal stabilizer (the tail). An alternative solution is the canard.

For chevron shaped (V-shaped) flying wings and tailless planes the solution is to twist the wing (in regular planes it's called washout but flying wing designers call it "twist"). The twist causes the wing tips to have less angle of attack (sometimes negative) so that the overall rotation force for the entire aircraft is zero.