If sin was the primary factor of physical death as many Christians believe, then death was not an option for Jesus. If Jesus was sin free, he'd be incapable of dying. That is a constant, because there was no sin found in him to die in any way. Jesus should be alive today, his body immortal and incapable of destruction.
But if Jesus choose to die and have his body damaged to the point of death, then does that mean it was sin that killed him.
Maybe our sins killed him, who knows.
You'll get a few different thoughts on it but I think most people who walk by the Spirit believe that what you've said is true, somewhat, that it was sin that brought death, because God said, on the day you eat of it, you will die.
But there are some divergences of opinion from there.
At first glance or first reading, you're left saying, but...they
didn't die on the day they ate it. In fact, they lived hundreds of years more. So you start wondering if it was some other kind of death. And you find reinforcement for this, (though you still kind of can't see what kind of death they died on that day) when you read that, technically, it wasn't sin that brought about their death but rather God said, look at what has happened --what if he now eats from the other tree and lives forever??
So God then blocks the way to that tree and they are not given the option to eat from that tree and live forever.
So technically, sin didn't cause them to not live forever, but being blocked from being allowed to eat of the tree that would cause them to live forever is what happened.
This is why I can't follow the reasoning of men who think we are born immortal and that we will all live forever. It doesn't mesh out. If Adam and Eve weren't created immortal, (and they weren't if they had to eat from the other tree to
become immortal), then how were the children they bore somehow immortal if they didn't eat from the tree either?