Y'know, I actually agree with you here. Even animal sacrifice was not efficacious without repentance (and in pertinent cases, restitution to the wronged), and it was only efficacious because God said that He would accept it for that purpose. The sacrifice was essentially symbolic, not efficacious in and of itself.
It doesn't matter that the Lord came into this world centuries after David died. The Lord still washed David.
This is the answer I typically get to this question. WhatNot's reply was interesting to me because it was taking a different angle. The fact that the passage from the psalm here is in the present (or past) tense doesn't pose any problem to you, even though the act of expiation wasn't made until 1000 years later?