As I understand conventional Christian doctrine, Christ's death on the cross atones for mankind's original sin (Adam & Eve, the forbidden fruit, etc.), and all subsequent sin...correct?
Ahh, but you stopped just short of answering the question. Why is such atonement necessary? What (or who) does it save us from? The point is Jesus is God saving us from himself. So in answer to your original question, yes God is very strict. So strict, he had to die to save us from his perfect rightousness.
Christ’s death on the cross - as the only man who never sinned - not even in thought was a substitionary sin for those who trust in God and no longer desire to be a slave to sin. The entire priesthood and sacrificial system in the Old Testament was to point to the work of Jesus Christ.
Perfect sacrifices were brought by individuals who desired their sins to be atoned for, they “transferred” their sin to the animal and a holy, (set apart) priest made the sacrifice. The priests were to be the mediators between the people and God and as such, they had to be set apart from the people. There were many cleansing rites to do that. One of them was to keep the altar burning using a specified fire. The sons of the high priest that were zapped with fire had rebelled against God in their hearts by doing what they wanted to do and not God.
Since sin is disobeying God, and the penalty for that is death. Physical death is seen more in the Old Testament, but we know that it also is a representation of spiritual death - which is separation from God. These sons were to be set apart as the emissaries, the ambassadors for God, but their hearts were shown to be wicked for doing things their way and not as they were told. And so, God, being just, had to zap them. They were no longer able to be priests even though by that system, they could not do anything but be a priest.
So, there was really no alternative. Having done less than zapping them would have compromised the whole sacrificial system, which in reality points to Christ Jesus. Sin is serious and not to be taken lightly. Worshipping God in the way that He has told us is a delight beyond anything that can be put into words. And the opposite it true - worshipping God in our way, turning our back on Him willfully is wicked beyond words. What the priests did was not just a sin of their own, but they were putting ALL the Israelites at risk. Sin is serious and so is the punishment for it.
The sacrificial system is about the grace and goodness of God reaching out to sinners who realize they cannot please Him in anything they do because they sin and all sin deserves death (separation from God). Only God Himself can provide a way to reach down and save them from that dilemma. And that is only by faith in that He is the only one who can. but His purpose was to teach the seriousness of willful sinning.
Christ Jesus, God Himself in the flesh, came to be our sacrifice and our priest. All of the sacrificial system points to that.