No, the Levitical system does NOT have this concept. That's your central problem. A small subset of sins can be atoned for by a process that includes a sacrifice. But not all sins, and the sacrifice need not be an animal of any sort.
All of the offering except the grain offering and he peace offering were for atonement. They taught that man did not have to die for his sins, a substituee could be offered fore his atonement. They also had a drink offering that was not drunk, it was poured out symbolizing death.
The grain offering is picture of the Lords' supper. Although no blood was present in it, to make the bread, the grain had to be crushed---the Lord was pleased to crush Him(Isa 53:10a). It had a "memorial portion(Lev 2:3)---This do in remembrance of me. The memorial portion was offered on the bronze altar, which is where the other offerings were placed. The rest of this offering was food for the priest and Christians are God's priests(I Pet 2:9).
The peace offering was for fellowship with God, not for sin. It was the only offering in which God, the priest and the offeror ate.
]ALL sacrifices had to be unblemished (no scourge marks...)
That is why Israel can't qualify as the suffering servant of Isas 53. That is why no mere human can qualify.
but no man dies for the sins of another.
Because no man is sinless. The sacrifice had to be without spot or blemish and only Jesus could meet that requirement.
Now, there is a concept within Judaism of a death bringing about the repentance of others and thus leading to atonement, but that's an advanced idea and you may not be ready for it.
If you have Scripture to support it, bring it on. If not don't bother
As for Ps. 14:3, you should read 14:4 and see who the "all" refers to and what their relationship is to the Jews.
Verse 4 does not have an * eliminating "no one" and "not even one," in verse 3.
Do you really think some people have never sinned?