Which saves are you targeting? If DX, it will miss about half. If Int, it will miss about 80%. Depends on caster stats of course--you said you have a Staff of Power, which ought to push failures up even higher.
You keep saying that legendary creatures always make their saves, but the math says otherwise. It depends which save you target.
BTW, why is your dragon only getting one legendary action per turn? It should be getting three, for a total DPR around 100 (times hit percentage) counting the lair action.
How does the Staff of Power push failures higher?
I did not say Legendary Creatures always make their saves. I said they also have good saves. Which means they make a lot of saves and then use Legendary Resistance for those few times they might miss and the spell is substantial enough to require it. For example, if you hit a dragon with a
fireball that you roll 27 points of damage on. The dragon misses its save. Depending on its hit points, it may just take the damage and keep the Legendary Resistance for something more dangerous. It only needs to use Legendary Resistance for effects that matter. You cannot trick it into using Legendary Resistance because it is a metagame ability.
You're not understanding the math. It is far more complex than "Attack the low stat." It involves more complex strategic concerns than the saving throw itself.
You either don't understand or don't want to accept that wasting an action casting a spell that isn't going to work for the sole purpose of making a creature use its Legendary Resistance is an advantage to the creature. You gave a creature with a DPR in excess of 50 a free round of actions, while you just wasted your time trying to make it burn something it doesn't need more than three of to kill you and your party. If we had all day and unlimited spell slots, maybe we could blow through its Legendary Resistance. But in actual play we have been unable to make a Legendary Creature use all of its Legendary Resistance during a battle for ten levels before it managed to beat us. So we discarded that strategy because it didn't work. Now we are looking for spells that provide maximum effectiveness without saves, so we don't die.
You seem to think we did not try to blow through its Legendary Resistance. We most certainly did try. The bard, cleric, and wizard coordinated spells to try to make it burn through it. Didn't work. After a couple of rounds of trying, well, our martials were near dead, the cleric was down, and the wizard and bard weren't doing well either. That meant we had to switch to spells that worked.
I'm advising you to not even try. Better to take spells that work like
scorching ray or similar spells because you want that dragon dead as soon as possible. If you have archer martials, you should be ok. Archers tear a dragon up. Dragon might as well run if your group is ranged heavy. it has next to no chance. You'll be able to add that much more damage with a ranged attack spell with no save. You'll be wasting time while your ranged archers kill it trying to beat Legendary Resistance.
Fights are fast and furious. Either it's dying or you're dying within three rounds.