Is it important that our justice system generally tries a defendant one time unless the defendant desires a retrial?
Certainly. If a defendant is acquitted, they still have spent considerable time and money in their defense that they cannot get back. They may never repair their reputation even if they are found not guilty. If the government could simply retry an individual as many times as they wished, many more innocent people could be harassed and deprived of property and freedom.
Double Jeopardy is prohibited under the United States constitution as are unreasonable searches and seizures; the same document guarantees every defendant a speedy public trial. All of these were based on abuses which the framers of the constitution perceived in the British legal system, or were rights preserved in the British system which they considered important. If a defendant were allowed to be tried more than once, there would be no end of prosecutions, and a defendant, whether guilty or innocent, would never know that the matter was finally behind him. This in itself is cruel.
Incidentally, in Australia, a defendant may be re-tried even if he has been found not guilty in certain instances, when one has perjured himself and the verdict might have been otherwise were it not for the perjury.
It is absolutely important that our justice system should have a prohibition on double jeopardy. A system without such a safeguard would be one in which the government could persecute people essentially in perpetuity without cause.
If we did not have a prohibition on double jeopardy, the government could retry people over and over again. This could essentially destroy a person's life. Repeated trials would bankrupt most people because of the costs of the attorneys and because a person who was constantly on trial would have a very hard time holding a job. The person's emotional life would also be devastated because they would be forever under the threat of trial.
A government that could try a person over and over would have essentially no limits on its power to destroy a person's life. This is why thinkers since the days of ancient Rome have argued that a prohibition on double jeopardy was absolutely essential for the rule of law.