Saves are reflexive reactions to a stimulus. They don't require actions, typically. They are automatic, and go at the beginning or end of your turn.
Checks require conscious effort, and are typically used to attempt to do something. So getting out of a grapple is a conscious effort.
As to whether Legendary Resistance would work, if I'm not mistaken it usually is worded as "If the creature fails the save, it can choose to succeed instead." I'd rule that if you want to be able to use that on checks, then sure. But it still takes an action, and he still has to fail for him to need to use the Legendary Resistance. He can just choose to succeed.
Action Taking Things are vastly more powerful to every type of foe. That's why they're so popular.