Sending the prophets and the Old Testament scribes to tell us about His love, sending His only Son into the world to teach us directly about God and His love and ultimately to die for our sins, filling our lives with His Blessed Sacraments, graces and the guidance of the Holy Spirit, THIS is washing His hands clean of any responsibility?
Yes, because you would say that none of those things directly resulted in someone going into Heaven. The things you list are just "helps" in Catholicism. If I walk up to my buddy, who is working on his car, and I hand him a wrench, I do not take any responsibility for whether the car gets fixed or not. I see Catholicism like that. God will hand you a wrench, but the rest is all on you. That means that God will hand a wrench to anyone, but it doesn't matter to Him which are able to fix the car and which cannot. He just stands there and watches.
If you see Catholicism like that, you have a very warped view of Catholicism.
Why do you say that it doesn't matter to Him? I am quite certain that He is big enough that EVERYONE matters to Him, even those who reject Him. After all, not even a sparrow falls without His knowing. You have said something similar several times. Do you not know that Christ died even for the sake of salvation for those who killed Him? "Forgive them, Father, for they know not what they do."
That is one of the merits of wearing and displaying the crucifix. It reminds us how much He loves us, each of us, even those of us who nailed him to the cross. That much.