Vdc in ripple factor
In short, why is V_dc the difference between the peak rectified unfiltered voltage and the peak-to-peak filtered voltage?


That's not what $$  V_{dc} = V_{p(rect)} - \frac {1}{2} V_{r(pp)} $$

is saying. It is saying that "Vdc is the peak rectified voltage and the peak minus half the ripple". This is clear from the graph.

In fact, and it is visible in the graph as drawn, this is an approximation. The area between the blue line and the average when above the blue line is greater than when below.