Something that plagues the concept of Str/Int/Cha saving throws is that these 3 abilities work mostly with an active roll: they typically represent taking a physical action, mental action, and exerting your force of personality.
Constitution and Wisdom work a lot more passively, and are more naturally suited to saving throws and reactions than actions (in fact, Constitution has no skills, and Wisdom skills work mostly reactively).
Dexterity is still about physical actions, but it gets a lot of saving throws, because dodging stuff works reactively.
So this is why IMHO when a new spell or monster's ability is designed, the designers ask themselves "how is the target going to try avoid/lessen/free herself from the effect?" and most of the time the natural suggestion is dodge (Dex), physically resist (Con) or mentally resist (Wis).
But I also believe that there is room for Str/Int/Cha saving throw, as long as the designer focus more on the fact that you have to (re)act fast or the effect will overtake you.
So all paralysis effects including petrification would work perfectly with a Strength saving throw: it's still a sort of action you have to take (which is kind of forcing your body to move, against the external force trying to make you stand still) but you have to take it immediately or you'll be turned to stone.
Charisma isn't even always that distinct from Wisdom. "Force of personality" vs "force of will" are conceptually a bit overlapping. Maybe the first is typically directed towards other people, while the second is typically directed towards yourself. But still, you could say you resist a compulsion with Wisdom ("I... will... do what I want!") but just as well you can say it works with Charisma ("You... won't... make me do what you want!").
Intelligence is the trickiest one, especially because often using intelligence is a long process. It's a bit harder to find good examples of fast, reactive uses of intelligence in the context of spells effects. Recalling information, make some quick calculation or reasoning yes, but still there aren't probably many spells effects traditionally in D&D. However, given that Illusions are given the chance to be identified in a matter of seconds with an Int(Investigation) check, they could have been done just as well with an Int saving throw.