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If you, dear Alex, die tomorrow, Christ's sacrifice would be sufficient for your salvation, yet some "purification" like the purification that Temple priests did would be needed before going into the presence of God.

Allow me to parse that for better understanding. In Catholicism:

"Salvation" does not mean "Presence of God"
Christ's sacrifice only paid for Salvation. His shed blood is an insufficient payment for entering (directly) into the Presence of God.
Before entering into the Presence of God, a "purification period" is required after death.
There is no full and sufficient payment that can be made, by man or by another on his behalf, to accomplish instant purification.
To achieve purification, an individual man must make his own "sufficient payment" via personal obedience, penance, and indulgence before death, or via time in Purgatory after death.

Did I get anything wrong?

I'm sorry, but your parsing is completely wrong.

In Christianity

"Salvation" means Salvation -- the general meaning of liberation from straitened circumstances or from other evils, and of a translation into a state of freedom and security. If you are in a state of purgatory, you ARE in a state of freedom and security because you ARE going to heaven, you're not going to hell. Purgatory is no more than washing your hands before entering the temple of God - metaphorically speaking
Christ's sacrifice paid for salvation. His shed blood is sufficient payment for us to enter into the presence of God. Remember the example of washing your hands before entering the Lord's presence -- THAT is the state of purgatory
li>Before entering into the Presence of God you need to be pure -- for the same reason that Aaronic priests had a cleanliness ritual, you too need to be purified to be able to enter into God's presence. Sin has no place in the presence of God and is burned away, hence purgatory is the process by which your sin is burnt away
There is full and sufficient payment for your salvation and entrance into heaven -- that's what Christ's sacrifice was about
To achieve purification,an individual man is cleansed by God -- THAT's what the purgatory process is about

It's really quite simple -- take the simple analogy of Aaronic priests washing themselves before entering the presence of God. Did the washing mean that God did not welcome them in? Or was washing the "payment" to enter in? NEITHER. Washing was the way to be pure.

in the same way, the process or state of purgatory is the metaphorical washing away of our sins to be able to bear the holiness that is God.

Does God do this completely and utterly? Yes.

Can we do anything to pay for this? NO. God (Jesus Christ) already paid

Purgatory is NOT a final test -- those who are in the state or process of purgatory are already not going to hell, their "ticket" has been paid for by Christ's sacrifice which is super-sufficient for your salvation.