If you were tried for murder and acquitted - then you go out and publicly admit that they were wrong, you did actually murder that person. Where is the justice in not being able to be tried again in those circumstances? There is now no doubt that you were in fact guilty - why tie the court/police's hands?
For example, the case of Emmett Till, where the murderers freely admitted that they did it after the fact, and could not be tried again.
In the United States, prohibition against double jeopardy is a constitutional protection. As long as one was actually at jeopardy for an offence by a particular sovereign, that sovereign may not subsequently prosecute the accused for the same wrong. In other jurisdictions, such as the U.K., it has a less strict form, even though generally, special pleas of autrefois acquit would be available.
It is also not the case that after an apparent "confession" in public that there is "no doubt" about a person's guilt. No evidence is "certain" in law without being tested in court.
Also, you propose:
you were tried for murder and acquitted - then you go out and publicly admit that they were wrong, you did actually murder that person
This does not put the acquittal into question. The prosecution failed to prove the case beyond a reasonable doubt. This means at law that one is deemed to be not guilty. Your scenario does not show that the trier of fact was "wrong" about the prosecution failing to prove its case beyond a reasonable doubt. This is not an avenue by which the prosecution can challenge an acquittal. If a properly instructed trier of fact finds that that the prosecution did not prove its case beyond a reasonable doubt, this conclusion is unassailable.
You ask "where is the justice?" The justice of a system is assessed by its application across the totality of cases. Given that a system of prosecuting and judging that is run through humans will inevitably produce errors, the law has developed to promote a measure of "justice" across the entirety of the cases that are disposed of by the court rather than to futilely attempt to ensure the "correct" result in every particular case.
The rule against double jeopardy has arisen out of this systemic concern for justice. It does not purport to secure the "correct" result in every case.
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.
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.
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.
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.
In the U.S., the bar against double jeopardy exists because court cases cost money both to the defendant and the tax payer. Since it is the state that can bring criminal cases and the state that has superior resources in evidence gathering, the law was implemented to prevent the state from ruining an innocent person by repeatedly trying them in court at the tax payer's expense. Additionally, it is to prevent the state, having once failed to convince the jury, to retry with another jury and continue to do so until they get a verdict they like.
While there are occasions where Double Jeopardy does not attach, these are exceedingly rare (one of the few times it was done without declaring a mistrial was in a case where the judge in a bench trial was bribed to return a not guilty verdict. Because the defendant was never in a legal jeopardy situation, it was ruled he could be tried again for the same criminal event). Additionally, the State and Federal Government are separate sovereigns for the purposes of Double Jeopardy, and while the Federal Government does not normally try someone who is tried by the state regardless of outcome, it is certainly possible for them to do so as the rules preventing them are policy, not law.
Another advantage of double jeopardy protection, beyond what has been mentioned already, is that it helps the truth to come out after the trial is over. For example, the murderers of Emmett Till would likely not have freely admitted to the crime had the possibility of criminal penalties still existed. There are several cases where this provides historians information that otherwise would have remained hidden.
