You have put the finger on a key distinction: in Islam there is no such thing as vicarious or expiatory sacrifice. From the Jewish and Christian standpoint, God in his love allows us to die vicariously in the form of a sacrificial animal (the ram that replaced Isaac on the altar), or in the Christian case, Jesus himself. I wrote about this some years ago:
Before the Bible was written, the Babylonian hero Gilgamesh learned that his quest for immortality was futile. The demigods of Greece, mortals favored by Olympians, suffered a tedious sort of immortal life as stars, trees or rivers. The gods of the heathens are not in any case eternal, only immortal. They were born and they will die, like the Norse gods at the Ragnorak, and their vulnerability projects the people's presentiment of its own death. To whom, precisely, have the gods offered eternal life prior to the appearance of revealed religion? Eternal life and a deathless mortality are quite different things.
But what is it that God demands of us in response to our demand for eternal life? We know the answer ourselves. To partake of life in another world we first must detach ourselves from this world in order to desire the next. In plain language, we must sacrifice ourselves. There is no concept of immortality without some concept of sacrifice, not in any culture or in any religion. That is a demand shared by the Catholic bishops and the Kalahari Bushmen.
God's covenant with Abraham is unique and singular in world history. A single universal and eternal god makes an eternal pact with a mortal that can be fulfilled only if Abraham's tribe becomes an eternal people. But the price of this pact is self-sacrifice. That is an existential mortal act beyond all ethics, as Soren Kierkegaard tells us in Fear and Trembling. The sacraments of revealed religion are sublimated human sacrifice, for the revealed god in his love for humankind spares the victim, just as God provided a ram in place of the bound Isaac on Mount Moriah. Among Jews the covenant must be renewed in each male child through a substitute form of human sacrifice, namely circumcision.  Christians believe that a single human sacrifice spared the rest of humankind.
Jihad also is a form of human sacrifice. He who serves Allah so faithfully as to die in the violent propagation of Islam goes straight to paradise, there to enjoy virgins or raisins, depending on the translation. But Allah is not the revealed god of loving kindness, or agape, but - pace Benedict - a god of reason, that is, of cold calculation. Islam admits no expiatory sacrifice. Everyone must carry his own spear.
We are too comfortable, too clean, too squeamish, too modern to descend into the terrible space where birth, death and immortality are decided. We forget that we cannot have eternal life unless we are ready to give up this one - and this the Muslim knows only through what we should call the sacrament of jihad. Through jihad, the Muslim does almost precisely what the Christian does at the Lord's Supper. It is the sacrifice of Jesus that grants immortal life to all Christians, that is, those who become one with Jesus by eating his flesh and drinking his blood so that the sacrifice also is theirs, at least in Catholic terms. Protestants substitute empathy identification with the crucified Christ for the trans-substantiated blood and flesh of Jesus.
Christians believe that Jesus died on the cross to give all men eternal life, on condition that they take part in his sacrifice, either through the physical communion of the Catholic Church or the empathetic Communion of Protestantism. From a Muslim vantage point, the extreme of divine humility embodied in Jesus' sacrifice is beyond reason. Allah, by contrast, deals with those who submit to him after the calculation of an earthly despot. He demands that all Muslims sacrifice themselves by becoming warriors and, if necessary, laying their lives down in the perpetual war against the enemies of Islam.
These are parallel acts, in which different peoples do different things, in the service of different deities, but for the same reason: for eternal life.
Why is self-sacrifice always and everywhere the cost of eternal life? It is not because a vengeful and sanguineous God demands his due before issuing us a visa to heaven. Quite the contrary: we must sacrifice our earthly self, our attachment to the pleasures and petty victories of our short mortal life if we really are to gain the eternal life that we desire. The animal led to the altar, indeed Jesus on the cross, is ourselves: we die along with the sacrifice and yet live, by the grace of God. YHWH did not want Isaac to die, but without taking Abraham to Mount Moriah, Abraham himself could not have been transformed into the man desirous and deserving of immortal life. Jesus died and took upon him the sins of the world, in Christian terms, precisely so that a vicarious sacrifice would redeem those who come to him.
What distinguishes Allah from YHWH and (in Christian belief) his son Jesus is love. God gives Jews and Christians a path that their foot can tread, one that is not too hard for mortals, to secure the unobtainable, namely immortal life, as if by miracle. Out of love God gives the Torah to the Jews, not because God is a stickler for the execution of 613 commandments, but because it is a path upon which the Jew may sacrifice and yet live, and receive his portion of the World to Come. The most important sacrifice in Judaism is the Sabbath - "our offering of rest", says the congregation in the Sabbath prayers - a day of inactivity that acknowledges that the Earth is the Lord's. It is a sacrifice, as it were, of ego. In this framework, incidentally, it is pointless to distinguish Judaism as a "religion of works" as opposed to Christianity as a "religion of faith".
To Christians, God offers the vicarious participation in his sacrifice of himself through his only son.
That is Grace: a free gift by God to men such that they may obtain eternal life. By a miracle, the human soul responds to the offer of Grace with a leap, a leap away from the attachments that hold us to this world, and a foretaste of the World to Come.
There is no Grace in Islam, no miracle, no expiatory sacrifice, no expression of love for mankind such that each Muslim need not be a sacrifice. On the contrary, the concept of jihad, in which the congregation of Islam is also the army, states that every single Muslim must sacrifice himself personally. Jihad is the precise equivalent of the Lord's Supper in Christianity and the Jewish Sabbath, the defining expression of sacrifice that opens the prospect of eternity to the mortal believer. To ask Islam to become moderate, to reform, to become a peaceful religion of personal conscience is the precise equivalent of asking Catholics to abolish Mass.