You are conflating, I think, the manner and location of the shedding of the blood with the fact of the shedding of the blood. Put simply, Jesus was the perfect spotless Lamb of God whose blood is shed for the remission of sins; there was no requirement that his blood be shed on the Jewish altar.
Consider the archetype of this sacrifice in God's command to Abraham to sacrifice his son. That was to be valid, even though there was no temple or altar.
2 Then God said, “Take your son, your only son, whom you love — Isaac — and go to the region of Moriah. Sacrifice him there as a burnt offering on a mountain I will show you.”
13 Abraham looked up and there in a thicket he saw a ram caught by its horns. He went over and took the ram and sacrificed it as a burnt offering instead of his son. 14 So Abraham called that place The LORD Will Provide. And to this day it is said, “On the mountain of the LORD it will be provided.”
15 The angel of the LORD called to Abraham from heaven a second time 16 and said, “I swear by myself, declares the LORD, that because you have done this and have not withheld your son, your only son, 17 I will surely bless you and make your descendants as numerous as the stars in the sky and as the sand on the seashore. Your descendants will take possession of the cities of their enemies, 18 and through your offspring all nations on earth will be blessed, because you have obeyed me.”