Calculating Average Voltage

So, that's a rule that applies to perfect inductors with the implied property that Bad Things happen if the current goes to infinity.

This is a pretty good model for the real world of power supply design, where long before you get to infinite current you get to enough current so that the magic smoke leaks out of something on your board (usually a chip, but sometimes the inductor or the board itself).

So it's only a condition imposed by the inductor in the sense that your circuit is damaged (or at least malfunctions badly) if you don't make sure it happens.

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.