What gets wrong when I average AC voltage and current to get power?
You have made a false math assumption. You have caught from the wind "taking average and multiplication are distributive". That's not true as you have already found. You can check it with two voltage samples U1, U2 and two current samples I1, I2

The average power is (U1*I1 + U2*I2)/2. There's no way to reduce this to ((U1+U2)/2)*((I1+I2)/2) 

You must calculate P(t)=U(t)*I(t). That's the momentary power. The average power is the average of U(t)*I(t) calculated over the period of interest. With sinusoidal current and voltage we calculate the average over one cycle.