I don’t have any evidence that this is the reason the spacebar is used for page down, but back in the day when IBM was setting PC design standards (that still heavily influence the design today), the original IBM AT 84-key keyboard from 1981 (IIRC) did not have page up/down or dedicated arrow keys (they shared the number pad):
The standard 101-key keyboard didn’t come on the scene until 3-ish years later. There were competitors to IBM too, and strange alternatives all over the place, e.g. the MIT Space Cadet, which has no page up/down, and has the hand symbols that vaguely resemble arrows:
From the link above re: the hand keys (emphasis mine):
Thumbs- up, down, left, and right keys. It was never quite clear
whether these were for answering yes/no questions or for navigation.
In the groups that used this keyboard, there was a cultural aversion
to arrow keys. To be honest, I’m not sure why any more.
The rapid development of new (and sometimes competing) technologies and interfaces at the time probably led to years of page up/down and arrows being an unreliable / inconsistent source of interaction (and standardization was not as wide-spread at the time), but one key all these keyboards had in common was the spacebar; it has always been around and unchanged.
Also, the mouse didn’t really fall into general consumer use until around the same time as the 101-key keyboard; with the first Mac’s in 1984, and it raised a lot of eyebrows. It was a while before the mouse was as much of a staple as it is today (and even today a mouse is still a bit of an option; you are far more likely to find a computer with no mouse than with no keyboard).
I imagine that this combined with the usage points jgthms brings up in his answer are the reasons that programs like
more (mentioned in Celada’s answer) and others used the spacebar so heavily —
more, for example, was written in 1978, well before any truly meaningful standardization of extra keyboard keys. Over time, it probably just stuck.