I think it is even better than that, screen savers were a hack that existed in the interstitial time between CRTs and LCDs because technology would not move fast enough.
The first CRTs were actually used as memory[1] and the fact that the phosphor 'remembered' what it had shown was an engineered thing. Then CRTs were used as displays that you could 'burn in' the contents was a problem [2].
That started the notion of not keeping the same thing on the screen all of the time which was "hard" to do for character driven terminal screens, but possible with bitmap driven screens. The first 'screen saver' I saw was at PARC on an Alto of all things which was drawing lissajous figures during idle time. They also appeared in arcades where screen burn was an issue so the game would display something else to avoid screen burn.
And THEN the energy crisis hit and everyone was looking at saving energy so CRT manufacturers added a simple circuit that if you disabled vertical sync they would detect that and actually turn off the electron beam completely. (horizontal sync was still needed to maintain the high voltage field at the front of the CRT). But by now a lot of people had grown attached to screen savers so turning off the screen was avoided.
And then we got to LCDs where running a screen saver was actively bad for the screen since there was no way to turn off the LCD backlight which was the weak point of the LCD screen and running a screen saver put more hours on that resulting in a faster death of your LCD.
[1] https://en.wikipedia.org/wiki/Williams_tube
[2] https://en.wikipedia.org/wiki/Screen_burn-in