Just because I believe I’m guilty doesn’t mean I am
All crimes contain elements, each of which must be true for someone to be guilty. Yes, I shot the victim, yes, I intended to kill him, yes, he’s now dead, yes, I believe I’m guilty. But if he died in his sleep just before I pulled the trigger, I’m not.
Confessions can be lies
I know it might shock you, but people are not always telling the truth when they confess to a crime. There are various reasons why someone might do this: mistaken belief, protecting someone else, status etc. Grotesque as it is, in the Emmett Till case, the acquitted men would have received a great deal of status in their community if they had committed the crime - that could be enough incentive for them to falsely confess.
Conflicting objects of the legal system
Justice is only one of the purposes of our legal system. Others include efficiency and finality.
Society can have only as much justice as it can afford - in both time and treasure. Yes, it would be more just if every case reached the correct result but it’s just too damn expensive.
Similarly, the legal process must come to a final and definitive end. It is neither efficient nor just for the process to be never ending.
Finally, in a criminal matter, compared to any given individual, the state has near limitless resources. It’s important not to allow the justice system to become a tool of persecution. You can’t allow the state to keep feeding coins into the slot machine until they hit the jackpot of a jury that agrees with them.
Some jurisdictions do allow retrials in certain situations
The rule against double jeopardy in the USA is a constitutional protection and, unless that changes, it is absolutely prohibited.
Other common law jurisdictions such as the UK and some Australian states allow the rule to be broken for serious crimes (like murder) where there is “fresh and compelling evidence”. An unsworn confession by the accused isn’t very compelling.