I think saves were originally more akin to hp than AC. Hit points scaled with level and stood between a character and death; saving throws scaled with level and stood between a character and death, paralysis, petrification, poison, and a bunch of other effects.
This is exactly correct.
In wargames (such as Chainmail) it used to be that one hit would kill. In fact, that's exactly how Chainmail worked- you get hit, you get killed. So a saving throw was used to avoid that result.
Assumedly, when Chainmail was adapted to Arneson's early attempts to create an RPG, he would allow the characters to survive hits by rolling to "save" (saving throw) their characters.
But you can see how that would still be ... kinda sucky. That's where we get armor class (making it harder to hit) and hit points (survivability to hits).
So by the time Arneson collaborated with Gygax, there were three methods (the alternative combat system) that allowed characters to survive longer than the old wargaming 'one hit and you're dead'- hit points, armor class, and saving throws.