
The button would just turn off the PSU, leaving the CPU and the disk drives etc.

There would be no popup asking you whether you really wanted to turn off power, and indeed typically no way for the OS to even perform a controlled shutdown. Thus, if you walked up to a computer that you assumed was turned off (because the screen was dark) and pressed the power button to turn it on, you would instead turn the computer off immediately, potentially losing any unsaved work. Purpose #1 one used to be a lot more important than one might assume from a modern perspective, in large part because up until the mid-to-late 90s most computers had a mechanical power switch with no software protection. They showed that the computer was turned on and functional. However, the graphical effects displayed by screensavers had two important purposes:

To elaborate on Jon Custer's comment: yes, for protecting the display against burn-in, a black screen would do just as well or better as any "screensaver".
