The heat comes from enrgy conversion. In the case where the capacitor is the source of energy (already charged), it does no work and converts no energy. The energy is converted by the resistor, not the capacitor. Of course we are discussing a theoretical (aka ideal) capacitor where there are no losses involved.
Quite correct. But when the capacitor "powers" the source, it is only returning energy produced by the source. It does no work or conversion to perform this function.
I believe the real advantage is in the understanding of the concept of apparent, real, and reactive power (also the terminology and units of measure). IMO, that is what the OP asked some 370 plus posts ago.
It is common to see the formula
W = E ? I ?
pf. Yet very few common texts provide a treatise on the subject. With all the rhetoric flying around in this thread, it is no wonder