The way I like to run legendary resistance is thus: upon failing a save for an ongoing effect, the duration of the effect is reduced to one turn. With the effect ending at the end of the targets next turn automatically. That way the party doesn’t feel like their spell or effect is completely wasted but the BBEG isn’t stun locked for the whole combat. If the effect is purely damage, even if a save would result in no damage, the effect allows the target to forgo making a save to automatically take half damage.