I thought the point was predicting a stall by knowing the approach speed will be so low that you will stall, not detecting a stall just before it happens

You can't control the weather, if there's a micro burst during landing you are in trouble.

GA planes have a stall warning horn (based on AoA). They just generally don't have a AoA indicator (though that might be a good idea indeed), relying on indicated airspeed instead, which (for given airplane mass) has a one-to-one [1] mapping to AoA in unaccelerated flight. That's why the concept of stall speed exists.

That is the case since, in unaccelerated flight, we need weight == lift, so

   W := m g == L := 1/2 rho v^2 c_L S

with m = mass, g = earth gravitational acceleration, 1/2 rho v^2 = fluid dynamic pressure which is measured by the pitot tube and displayed as indicated airspeed (well, a function of it), c_L = the coefficient of lift, and S = wing area.

Now, weight is constant (for given airplane mass, in unaccelerated flight), and so is the wing area. The coefficient of lift depends on the AoA, and dynamic pressure has a monotonic one-to-one relationship to IAS. Thus you have the relationship between IAS and AoA.

[1] Unless you get to "the back of the power curve" (the coefficient of lift increases with AoA, then decreases again, until it drops off in a stall). Let's not go there.