I have a question: why blood (sacrifices) at all? – since G-d forgives whenever people repent (teshuvah). There are numerous examples in the Tanach which proves that G-d forgave people without them having to bring a sacrifice.
It seems that teshuvah is the condition, necessary and sufficient meanwhile.
You are a careful reader of Tanakh - very good!
In fact, God does indeed forgive people without blood sacrifices. The sacrifices described in Leviticus were given to us because of our need to make sacrifices; God never needed them. With the destruction of the Temple in the year 70, we moved to a better kind of sacrifice, sacrifices of prayer. Today we pray on the same schedule that once was the schedule for the sacrifices. That is why we say the Amidah prayer several times a day; it is our "sacrifice."
This is also the reason that we in the Reform Movement of Judaism do not look for a rebuilding of the Temple. We believe its work is complete; there is no need for blood sacrifices today.
And yes, teshuvah is what we need in order to be forgiven. The prayers within the Amidah move us towards teshuvah. In fact, one of them (in the weekday prayer) is very specific about teshuvah, but the other prayers in the great Prayer work to make our hearts more inclined to make teshuvah, to turn to goodness.
Thank you for an excellent question. L'shalom, Rabbi Ruth Adar