I would find it really strange if after 4600 years God suddenly changed his mind from sacrifices to having to only rely on your own good deeds.
Well on that, for the mode of salvation to change at all between the creation of the universe to the day of judgement would be pretty strange. One would think that God would stay consistent for all people.
On soteriology, it is expected for Christians to misrepresent the theology of the ancient Israelites and Judaism via particularly the writings of Paul.
Animal sacrifices are easy to do you just need to breed the animals to sacrifice, it was a very common practice in the ancient world.
The Ancient Israelites practiced it just as their neighboring nations did. In the Torah and the Tanakh (Old Testament) as a whole, animal sacrifice is
NOT the mode of salvation, repentance is.
In both Judaism and in Islam, a person atones and repents for their own sins. It is only through God directly.
Christianity on the other end has this very fanatical idea of a single substitutional human sacrifice which abrogates self-atonement. The function of this substitutional sacrifice remains a matter of incoherence, as a substitutional sacrifice realistically can only be situational (and limited to a specific historical context) and not universal. Whereas self-atonement of a believer to God is universal.
On sacrifice more broadly, it's a pretty deep thing in our subconscious. We naturally have an irrational impulse to sacrifice things. Over history this has "reformed" to becoming taken as a poetic metaphor rather than literal sacrifice.
Instead of sacrificing your cattle, you now "sacrifice" your time or your money or your possessions, etc towards a perceived greater good.
In the same way that Christianity used Jesus to do this to abrogate the Jewish animal sacrifice practices, so too did the Hindus nominally do this in their own reforms where they turned from animal sacrifice towards developing systems of meditation and yoga.
The Greeks were also gravitating towards related thought too regarding their sacrificial practices and then growing to metaphoricalize it.
I find this process throughout history, which Christianity follows suite with (rather than going against), to be intriguing. To show what I mean, if I say the words "animal sacrifice", you'll probably get squeamish. It's quite ridiculous.
Anyway in ancient Israel and in Judaism, animal sacrifice was a very popular practice but it was never the mode of salvation, repentance was. Repentance is in Islam likewise.
To show how inescapable it is, when you peel apart all of the fluffy language and overexaggerated symbiology, Christianity has the same basic mode of salvation only dressed up more convoluted.
As for Jesus's own words on this matter,
Matthew 5:17-20. Contrary to what you will staunchly continue to believe, Jesus didn't abrogate anything, he denied it explicitly.
Since you mentioned Islam, again, this verse is quite pertinent:
Qur'an, Surah 2:177
Righteousness is not to turn your faces to the east or the west;
rather, righteousness is [personified by] those who have faith in God and the Last Day,
the angels, the Book, and the prophets, and who give their wealth, for the love of Him,
to relatives, orphans, the needy, the traveller and the beggar,
and for [the freeing of] the slaves, and maintain the prayer and give the zakāt,
and those who fulfill their covenants, when they pledge themselves,
and those who are patient in stress and distress,
and in the heat of battle. They are the ones who are true [to their covenant],
and it is they who are the Godwary.
