The essence of sacrifice is giving up something dear to demonstrate your devotion to God. The Crucifixion cannot be explained as a sacrifice, because Jesus was rejected by the Jewish authorities and handed-over to the pagan Romans to be executed. The Crucifixion is more like the story of Joseph being sold into slavery by his angry brothers and then rising with God's help to a high office where he could repay his brothers with love and mercy in their hour of need.
No, sacrifice is voluntarily dying to achieve something important. We talk about soldiers making sacrifices. Vicarious sacrifice is dying in someone else's place. That's what the Jewish examples I cited were. You're thinking of animal sacrifice, where an offering is given to God. The concept of sacrifice is a lot broader than that.
And even among that kind of cultic sacrifice, there are different types. In the OT we have sacrifices for sin, as part of establishing a covenant, and fellowship sacrifices, which are simply normal worship.
The words of institution are “This is my body that is for you. Do this in remembrance of me.” “This cup is the new covenant in my blood. Do this, as often as you drink it, in remembrance of me.” Mark's version of the second: “This is my blood of the covenant, which is poured out for many."
This is sacrificial. It indicates he is dying for his people. But more specifically, Paul's version paraphrases and Mark's quotes Ex 24:8, thus making it clear that it is intended as a covenant sacrifice. The covenant would be Jer 31:31
What is it not is a sin sacrifice.
I've thought about what Jesus might have had in mind. First, the Gospels say he went to Jerusalem knowing what was going to happen. That is plausible. We know that he thought his people were headed for disaster. While there's no explicit explanation like this in the text, I think given the Jewish traditions, it is plausible to think he intended it as a vicarious act of repentance on behalf of his people.
What did Paul think the significance was? The only explicit explanation is Rom 6. You should read the text carefully. That is a sacrifice in the sense that it's a death for our benefit. He does also quote the words of institution though, which I think are sacrificial in a slightly different sense. Phil 2:8 sees it as an act of obedience. I would call it a sacrifice in the sense that he submitted voluntarily for our benefit. Rom 3:25 speaks of it as a sacrifice. Just what kind is disputed. It uses a word that alludes to the OT sacrificial altar, but may not specify a particular way in which it worked. It's sometimes translated propitiation, but sacrifice of atonement seems safer, since propitiation has implications that are probably more specific than the language justifies.
Now we get beyond the undisputed letters. Eph 2:13 says he made peace through his death, though it isn't explicit how. Col 1:20 is the same. Col 2:14 talks about nailing the law to the cross, suggesting perhaps that by being condemned according to the Law and then vindicated by God, it ends the force of the Law, though a substitutionary understanding would also work.
Sacrifice is very clear. Certainly in Paul. But I think with Jesus as well. The fact that he accepted death voluntarily is hard to dispute. That makes it a sacrifice in the broadest sense. If the Words of Institution are accurate it's even a cultic sacrifice. Given that these are present in both the Synoptics and Paul the words are very old. I'm going to see they go back to Jesus, but that can always be disputed. What's not present is penal substitution, at least in the sense the CF writers normally mean it.
