Double Jeopardy does not undermine the justice system. Double jeopardy is needed to keep legal matters from dragging on for lifetimes. It is a fair rule as it gives the prosecution a shot at the accused and allows the accused to go on with their life if they are not found guilty.
The double jeopardy rule does not allow a person to be charged for the same crime twice. This does not undermine the law in any way shape or form. The double jeopardy rule is a good thing. If we did not have it, civilians would regularly be harassed and recharged for the same crimes over and over again. Our court system is backed up enough as it is. It would be worse if we were wasting our time retrying old cases with people who had already been proven not guilty.