Yes, in that case (because that was where you brought us).Well actually, it does say that the blood makes atonement for one's life. What it doesn't say is that the altar makes atonement for one's life. It appears from v.7 that the Israelites had been making sacrifices that were not specifically to the LORD but, apparently, to false gods and demons. Therefore God commanded them to bring their sacrifice to the door of the Tabernacle and for the priests to offer the blood upon the altar.
However, when David was bringing the Ark up to Jerusalem, it seems that every six paces that he travelled, he sacrificed oxen and sheep (2 Sam. 6:13), yet the Tabernacle was in the city (v.17).
But it is this way in every case. In atoning for the altar (cleansing) the blood is applied. The door of meeting? It is the blood applied. Same with covenants. It is not the shedding of blood but the sprinkling.
But you are right - these are not directly speaking of atoning for sins.
So let's look at the Day of Atonement.
“He shall then slaughter the goat for the sin offering for the people and take its blood behind the curtain and do with it as he did with the bull’s blood: He shall sprinkle it on the atonement cover and in front of it. 16 In this way he will make atonement for the Most Holy Place because of the uncleanness and rebellion of the Israelites, whatever their sins have been. He is to do the same for the tent of meeting, which is among them in the midst of their uncleanness. 17 No one is to be in the tent of meeting from the time Aaron goes in to make atonement in the Most Holy Place until he comes out, having made atonement for himself, his household and the whole community of Israel.:
Lets look at the Passover as well.
Did death pass over the families when God noticed they had shed the blood of a lamb?
Nope. It was their application of the blood.
Taking life (removing life from the animal) is mot, as PSA theorists assume, the atonement according to Scriptute.
It was always the application of the blood (the application of this life) that was viewed as making atonement.
Yes, without the shedding of blood there is no forgiveness.
But you are wrong to stop there.