To accept these premise, we need to assume that someone thought that inside the sensor, light was replicated and grew like a seed into more light, out of nowhere. How I came to understand the ISO, and how most will understand it based on how it's explained, is that it take less light but produce a similar amount of lightness in the final scene, with more noise or less cleanness than otherwise.
The intuitive sense is when we turn the volume up. It's obvious we are scaling, and that amplification doesn't increase the fidelity, we just make it louder, amplifying the noise and the signal with it. Anyone that cranked the volume way up will note this. And anyone with any basic understanding will know the amplifier doesn't make a bad recording sound better.
I think the most important miss-understanding, looking into detail is...does anything happen before the signal is amplified digitally? Not much, there is some gain that happens or kicks in in some cameras and increases the DR. Most of the work is done digitally, including adjusting the algorithms to render the scene in ways that do not show significant color casts when sample rate goes down (ie ISO goes up).
That's so far, a summary of my current understanding. And usually the choice is...do I want more noise in form of blur, or in form of low samples? Current cell phones get away with, in part, thanks to their meagre sensors, very fast readouts. So they get to do very quick reads (most without blur, but if one has a lot, they can discard that one) and thus in a sense, get something that performs much better than what the sensor would show for one exposure.
If you set a high ISO in M and bombard the sensor, recording RAW....nothing bad happens if the amount of light is already adquate for the sensor. If you use JPG, the lightness gets cooked into the file and everything is blown without much fix, as the detail is already lost.
Most of the confusion is about what the multiple settings in the camera do together. For example, I could have spot metering and aim into a bright source, which will overexpose everything except that spot, but also set ISO 25000, and record raw, and maybe it's just the same as a well exposed matrix photo with ISO 100. To understand why, instead of the mistery of ISO, one only needs to understand the concepts of metering, the Mode, ISO (the setting not the concept), and how sensor works.
