Am I reading those rules correctly?
No. You can apply Quivering Palm with any strike that hits by expending 3 ki; but it costs a full action to trigger the death effect, and only then does the target roll its saving throw. Thus, it will take you a minimum of 8 rounds and 12 ki to kill the tarrasque this way.
Round 1: Attack, set up Quivering Palm.
Round 2: Trigger Quivering Palm. Tarrasque fails saving throw, uses Legendary Resistance.
Round 3: Attack, set up Quivering Palm.
Round 4: Trigger Quivering Palm. Tarrasque fails saving throw, uses Legendary Resistance.
Round 5: Attack, set up Quivering Palm.
Round 6: Trigger Quivering Palm. Tarrasque fails saving throw, uses Legendary Resistance.
Round 7: Attack, set up Quivering Palm.
Round 8: Trigger Quivering Palm. Tarrasque fails saving throw, has no more Legendary Resistance, dies.
And that's assuming the tarrasque fails every save. Remember that the tarrasque can wait to use Legendary Resistance until it sees the result of its saving throw; if it makes the save, it doesn't expend LR. With +10 Con save against the monk's DC of 19, it has a 60% chance to succeed, so on average it's going to take you 10 tries to bring it down (each attempt results in an average 0.4 failed saves, and you need a total of 4 failed saves to kill). That's 20 rounds. Not only that, but it's 30 ki, which you ain't got.
Bottom line: Even if you somehow manage to get the attack and the death trigger on the same round, and find a way to make the tarrasque fail all its saving throws, that's still 4 rounds you have to survive going toe-to-toe with the tarrasque in melee. Good luck.