Sorry,I misunderstood your point.
I also stuffed up with "OK,things get messy in reactive circuits,but P=VI was derived from a purely resistive case,& there are many situations where we can assume a resistive load."
What I should have said was that P=VI was derived from the purely resistive DC case,with P=I^2R,& V^2/R,further derived from that.(those are the "derived" formulae I was referring to)
Irms & Vrms were further determined by calculation & experiment.
And,yes,I am aware of the use of rms in other fields---I used it to find the area under the curve in Mathematics.
RMS current & voltage have been used extensively for over 100years,so they do still have their place.
In the case of a Radio Transmitter,a substantially resistive Test Load is used to determine the RF output power.
This can be,& is,often done using P=I^2R.or V^2R with lower power equipment.
(Larger installations often use a water cooled Test load,with average power determined by flow rate & temperature rise)
A real antenna is then matched to the Transmitter,so that it looks as near to resistive as possible,so the Transmitter can deliver the same power output as to the Test Load.
The incandescent lightbulb case really only applies during the initial warm up period--once it is hot,it is a pretty stable resistance.
My original comments were a bit "dumbed down" to make the point about so-called RMS Power.
I never intended to get into an esoteric "off topic" conversation, so if you don't mind, I'm going to 'bail" on it.