What people have a problem with is that you insist on calling it "power gain." It's not power gain. It's energy gain.
Average power out over some specified interval doesn't "just happen" to be equal to energy. It IS energy.
The above is like saying that miles per hour is a unit of distance. The *mile* (with no mention of time) is a unit of distance. Once you start talking about miles traveled over some specified interval, (MPH) you are no longer talking about distance (saying that MPH is a measure of distance that "just happens" to equal speed doesn't make it so). You are now talking about velocity -- distance traveled over a specified interval. Speed is not Distance. Distance is not Speed. Power is not Energy. Energy is not Power. One involves time. The other does not. You cannot average power over time and still get power, just as you cannot average distance over time and still get distance. Trying to do that leads to mathematical nonsense such as saying that the distance from the hellhole that is California to the utopia which is Arizona is infinite if you wait long enough, or that the power gain of a latching relay is infinite if you wait long enough. Both are meaningless because distance and power are unrelated to time, while speed and energy are. This has been explained to you multiple times.