To avoid some confusion you might rephrase "An analog white noise zapper doesn't have a fundamental frequency and a few harmonics," to "An analog white noise zapper doesn't have just a fundamental frequency and just a few harmonics,"?
The principle that you offer is correct if the real world battery could accomplish what you suggest, but it can not. That is why I suggest that it the white noise generator produces 1 million frequencies, then the average output of these frequencies is going to be approximately 1 millionth of the total power available from the battery source.
Accepting that there can be a variation in the power at any specific frequency and this variation is found to be +/- 2000 percent ( 20 times base ), then the highest energy frequency output would be 20/1000000 of the total power, which is still miniscule compared to the 41st harmonic of 2500 or even the 79th harmonic of 2500. In addition, this peak power is random, not repetitive as defined by white noise.
I absolutely agree that white noise is important because the universe itself is likely a white noise generator and white noise has long been known to have positive benefits. However, when certain specific frequencies are desired, the 50 percent duty cycle square wave is the best way of producing that, excluding having a pure sine wave of the right frequency and power.
I also subject that the sharpness of the leading and trailing edge of the square wave is actually significant because that is similar to a tap that starts ringing at any and all frequencies. This ringing is then supported by the harmonics present, and reinforced by the exact synchronous falling or leading edge. I subject that it is this combination that produces the maximum desired results.